In a system of N particles located in a Cartesian coordinate system, we can show that the Lagrange equations of motion reduce to Newton's equations of motion. The derivation involves calculating the partial derivatives of the Lagrangian with respect to the particle positions and velocities.
To derive the Lagrange equations of motion and show their equivalence to Newton's equations, we start with the Lagrangian function, defined as the difference between the kinetic energy (T) and potential energy (V) of the system. The Lagrangian is given by L = T - V.
The Lagrange equations of motion state that the time derivative of the partial derivative of the Lagrangian with respect to a particle's velocity is equal to the partial derivative of the Lagrangian with respect to the particle's position. Mathematically, it can be written as d/dt (∂L/∂(dx/dt)) = ∂L/∂x.
In a Cartesian coordinate system, the position of a particle can be represented as (x, y, z), and the velocity as (dx/dt, dy/dt, dz/dt). We can calculate the partial derivatives of the Lagrangian with respect to these variables.
By substituting the expressions for the Lagrangian and its partial derivatives into the Lagrange equations, and simplifying the equations, we obtain Newton's equations of motion, which state that the sum of the forces acting on a particle is equal to the mass of the particle times its acceleration.
Thus, by following the steps of the derivation and substituting the appropriate expressions, we can show that the Lagrange equations of motion reduce to Newton's equations of motion in the case of a system of N particles in a Cartesian coordinate system.
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A 0.045 kg tennis ball travelling east at 15.5 m/s is struck by a tennis racquet, giving it a velocity of 26.3 m/s, west. What are the magnitude and direction of the impulse given to the ball? Define the magnitude and for direction if it is west, consider stating the negative sign, otherwise do not state it. Record your answer to two digits after the decimal point. No units Your Answer: Answer D Add attachments to support your work A 67.7 kg athlete steps off a h=13.3 m high platform and drops onto a trampoline. As the trampoline stretches, it brings him to a stop d=1.4 m above the ground. How much energy must have been momentarily stored in the trampoline when he came to rest? Hint: it is coming to rest at height d=1.4 m from the ground. Round your answer to two digits after the decimal point. No units Your Answer: Answer A stationary object explodes into two fragments. A 5.83 kg fragment moves westwards at 2.82 m/s. What are the kinetic energy of the remaining 3.24 kg fragment? Consider the sign convention: (E and N+ and W and S− ) Round your answer to two digits after the decimal point. No units Your Answer: Answer A 2180 kg vehicle travelling westward at 45.4 m/s is subjected to a 2.84×104 N⋅s impulse northward. What is the direction of the final momentum of the vehicle? State the angle with the horizontal axes Round your answer to two digits after the decimal point. No units Your Answer: Answer
1. Magnitude of the impulseThe initial momentum of the tennis ball is given bym1v1 = 0.045 kg × 15.5 m/s = 0.6975 kg·m/sThe final momentum of the tennis ball is given bym1v2 = 0.045 kg × (-26.3 m/s) = -1.1835 kg·m/sTherefore, the change in momentum is given byΔp = p2 - p1= (-1.1835) - (0.6975)= -1.881 kg·m/sThe magnitude of the impulse is the absolute value of the change in momentum, which is|Δp| = |-1.881| = 1.881 kg·m/s(rounded to two decimal places).
2. Direction of the impulseThe impulse is in the opposite direction to the change in momentum, which is westward. Therefore, the direction of the impulse is eastward.Note that if we use a positive sign convention for eastward and a negative sign convention for westward, then the direction of the impulse can be expressed as-1.881 J (eastward).
3. Stored energy on the trampolineThe athlete loses gravitational potential energy (GPE) when stepping off the platform. This energy is converted into elastic potential energy (EPE) as the trampoline stretches. Therefore,GPE = EPEGPE lost = mghwhere m is the mass of the athlete, g is the acceleration due to gravity, and h is the height of the platform above the ground.GPE lost = 67.7 kg × 9.8 m/s² × 13.3 m = 93506.62 JWhen the athlete is at the maximum height d above the ground, all of the GPE is converted into EPE. Therefore,EPE stored = GPE lost = 93506.62 JWhen the athlete comes to rest, all of the EPE is converted back into GPE. Therefore,GPE gained = EPE stored = 93506.62 JWhen the athlete is at a height of d = 1.4 m above the ground,GPE gained = mghGPE gained = 67.7 kg × 9.8 m/s² × 1.4 m = 929.012 JTherefore, the energy momentarily stored in the trampoline when the athlete came to rest was 929.012 J (rounded to two decimal places).
4. Kinetic energy of the remaining fragmentIf the initial kinetic energy of the object is K1 and the kinetic energy of one of the fragments is K2, thenK1 = K2 + K3where K3 is the kinetic energy of the other fragment.Since the object is stationary before the explosion, its initial kinetic energy is zero. Therefore,K2 + K3 = 0andK2 = - K3The kinetic energy of the remaining 3.24 kg fragment (K2) is given byK2 = (1/2) m2 v²where m2 is the mass of the remaining fragment, and v is its velocity.K2 = (1/2) × 3.24 kg × (2.82 m/s)²K2 = 10.8748 JTherefore, the kinetic energy of the remaining 3.24 kg fragment is 10.8748 J (rounded to two decimal places).
5. Direction of the final momentumThe initial momentum of the vehicle is given byp1 = m1v1where m1 is the mass of the vehicle, and v1 is its velocity.p1 = 2180 kg × (-45.4 m/s)p1 = -99172 kg·m/sThe impulse acting on the vehicle is given byJ = Δpp2 - p1 = (0, Jy, 0)where Jy is the y-component of the impulse. Since the impulse is northward, Jy is positive.The final momentum of the vehicle is given byp2 = p1 + Jp2 = (-99172, Jy, 0)The magnitude of the final momentum is given by|p2| = √(p²x + p²y + p²z)|p2| = √((-99172)² + J²).The direction of the final momentum is given by the angle θ between the final momentum and the horizontal axis, measured counterclockwise from the positive x-axis.tan(θ) = p2y / p2xθ = tan⁻¹(p2y / p2x)θ = tan⁻¹(Jy / (-99172))Therefore, the direction of the final momentum is (rounded to two decimal places).
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A truck with a mass of 1890 kg and moving with a speed of 14.5 m/s rear-ends a 791 kg car stopped at an intersection. The con i cortes neutral, the brakes are off, the metal bumpers line up well and do not get damaged. Find the speed of both vehicles afer the common in meter per cond car
Answer:
The speed of both vehicles after the collision is approximately 14.5 m/s.
Given:
Mass of the truck (m1) = 1890 kg
Mass of the car (m2) = 791 kg
Initial velocity of the truck (v1) = 14.5 m/s
Initial velocity of the car (v2) = 0 m/s (since it is stopped)
Let's denote the final velocity of the truck as v1' and the final velocity of the car as v2'.
Using the conservation of momentum, we can write:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
Plugging in the given values:
(1890 kg * 14.5 m/s) + (791 kg * 0 m/s)
= (1890 kg * v1') + (791 kg * v2')
27345 kg·m/s = 1890 kg * v1' + 0 kg·m/s
Now, we can solve for the final velocity of the truck (v1'):
1890 kg * v1' = 27345 kg·m/s
v1' = 27345 kg·m/s / 1890 kg
v1' ≈ 14.5 m/s
The final velocity of the truck (v1') after the collision is approximately 14.5 m/s.
Since the bumpers line up well and no external forces act on the system, the final velocity of the car (v2') will be equal to the final velocity of the truck:
v2' ≈ 14.5 m/s
Therefore, the speed of both vehicles after the collision is approximately 14.5 m/s.
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Buck - Boost converter system parameters: Vg=48V input voltage, output voltage Vo=12V, output load R=1~100Ω, output filter inductance L=100μH, capacitance C=220μF, switch frequency fsw=40kHz, namely switch cycle Tsw=25μs. PWM modulator sawtooth amplitude VM=2.5V. Feedback current network transfer function Hi(s)=1 feedback partial voltage network transfer function Hv(s)=0.5
Draw the circuit and give Detailed derivation of the transfer function.
The Buck-Boost converter system consists of an input voltage of 48V, an output voltage of 12V, and various parameters such as load resistance, filter inductance, capacitance, switch frequency, and PWM modulator sawtooth amplitude. The feedback current network transfer function is given as Hi(s) = 1, and the feedback partial voltage network transfer function is Hv(s) = 0.5. The circuit diagram and transfer function derivation will be explained in detail.
The Buck-Boost converter is a DC-DC power converter that can step up or step down the input voltage to achieve the desired output voltage. Here is a step-by-step explanation of the circuit and the derivation of the transfer function:
1. Circuit Diagram: The circuit consists of an input voltage source (Vg), an inductor (L), a switch (S), a diode (D), a capacitor (C), and the load resistance (R). The PWM modulator generates a sawtooth waveform (VM) used for switching control.
2. Operation: During the switch ON period, energy is stored in the inductor. During the switch OFF period, the stored energy is transferred to the output.
3. Transfer Function Derivation: To derive the transfer function, we analyze the circuit using small-signal linearized models and Laplace transforms.
4. Voltage Transfer Function: By applying Kirchhoff's voltage law and using the small-signal model, we can derive the voltage transfer function Vo(s)/Vg(s) as a function of the circuit components.
5. Current Transfer Function: Similarly, by analyzing the current flow in the circuit, we can derive the current transfer function Io(s)/Vg(s) as a function of the circuit components.
6. Feedback Transfer Functions: The given feedback transfer functions, Hi(s) and Hv(s), relate the feedback current and voltage to the input voltage.
7. Overall Transfer Function: The overall transfer function of the Buck-Boost converter system can be obtained by combining the voltage transfer function, current transfer function, and feedback transfer functions.
By following these steps, the detailed derivation of the transfer function for the Buck-Boost converter system can be obtained. The transfer function describes the relationship between the input voltage and the output voltage, and it helps in analyzing and designing the converter system for the desired performance.
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Two stationary point charges experience a mutual electric force of magnitude 108 N. Subsequently, if the distance between the two point charges is tripled while the magnitude of both charges is cut in half.
What is the magnitude of the resultant electric force on either charge?
a. 6.0 N
b. 3.0 N
c. 12 N
d. 9.0 N
e. 27 N
The correct answer the magnitude of the resultant electric force on either charge is Option d.9.0 N
Let the original magnitude of one charge be q1 and the original magnitude of the other charge be q2. The original distance between the two charges is r.
The magnitude of the force between two point charges q1 and q2 is given by Coulomb's law as:F=kq1q2/r²Where k is Coulomb's constant which is 9 × 10^9 Nm²/C².Subsequently, if the distance between the two point charges is tripled while the magnitude of both charges is cut in half, the new distance between the two charges is 3r and the new magnitude of both charges is (1/2)q.
The force between the two charges with the new conditions is given by:F'=k((1/2)q)(1/2)q/(3r)²F'=kq²/27r²Since the magnitude of the force between two stationary point charges is the same for each charge, the magnitude of the resultant electric force on either charge is given by:F''=F'/2F''=kq²/54r²The ratio of the new force to the old force is:F''/F=kq/108r².
The magnitude of the force on each charge is:F1=F2=F''/2F1=F2=kq²/108r²The magnitude of the force on each charge is kq²/108r². Answer: d. 9.0 N.
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What do you need to find the intensity of an electromagnetic wave?
Both the electric and magnetic field strengths.
Either the electric or magnetic field strength.
Only the electric field strength.
Only the magnetic field strength.
To find the intensity of an electromagnetic wave, we need to know the electric and magnetic field strengths as they are interdependent. The correct option is 1) Both the electric and magnetic field strengths.
The intensity of an electromagnetic wave is given by the energy transferred per unit area per unit time and is proportional to the square of the electric and magnetic field strengths. Therefore, if either the electric or magnetic field strength is missing, it will be impossible to determine the intensity accurately. The electric and magnetic fields oscillate perpendicular to each other and the direction of propagation of the wave. They have the same amplitude, frequency, and wavelength, but they differ in phase.
The intensity of an electromagnetic wave can also be determined by measuring the average power per unit area over a period. In summary, both electric and magnetic field strengths are required to calculate the intensity of an electromagnetic wave accurately. It is important to note that these fields are interdependent on each other, and a change in one can affect the other. Therefore, accurate measurements are crucial in the determination of the intensity of electromagnetic waves.
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Calculate the maximum kinetic energy of a beta particle when 19K decays via 3.
The Q-value of the decay is 21.46 MeV.The electron binding energy of 19Ca is 3.210 MeV. Therefore, the maximum kinetic energy of the beta particle is:Kmax = Q – EbKmax = 21.46 MeV – 3.210 MeVKmax = 18.25 MeV
When 19K decays to 19Ca via β− decay, the maximum kinetic energy of the beta particle can be calculated by using the following formula: Kmax = Q – Eb Here, Kmax is the maximum kinetic energy of the beta particle, Q is the Q-value of the decay, and Eb is the electron binding energy of the 19Ca atom.
The Q-value of the decay can be calculated using the mass-energy balance equation.
This equation is given by:m(19K)c² = m(19Ca)c² + melectronc² + QHere, melectronc² is the rest mass energy of the electron, which is equal to 0.511 MeV/c².
Substituting the atomic masses from the periodic table, we get:m(19K) = 18.998 403 163 u, m(19Ca) = 18.973 847 u.
Substituting these values into the equation and simplifying, we get:Q = [m(19K) – m(19Ca) – melectron]c²Q = [18.998 403 163 u – 18.973 847 u – 0.000 548 579 u] × (931.5 MeV/u)Q = 0.023 007 u × (931.5 MeV/u)Q = 21.46 MeV
Therefore, the Q-value of the decay is 21.46 MeV. The electron binding energy of 19Ca is 3.210 MeV. Therefore, the maximum kinetic energy of the beta particle is: Kmax = Q – EbKmax = 21.46 MeV – 3.210 MeVKmax = 18.25 MeV
Therefore, the maximum kinetic energy of the beta particle is 18.25 MeV.
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a) Calculate the wavelength of light emitted by a Hydrogen atom when its electron decays from the n=3 to the n=1 state energy level. b) With respect to the photoelectric effect, the work function of Lead ( Pb) is 4.25eV. What is the cut-off wavelength of Pb ? c) A sample of Pb is illuminated with light having the wavelength calculated in part a). Calculate the velocity of the emitted electrons.
a) When an electron in a hydrogen atom transitions from the n=3 to the n=1 energy level, the wavelength of light emitted can be calculated using the Rydberg formula: 1/λ = R_H * (1/n_1^2 - 1/n_2^2), where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately 1.097 × 10^7 m^-1), n_1 is the initial energy level (n=3), and n_2 is the final energy level (n=1).
b) The cut-off wavelength of lead (Pb) can be determined based on the work function, which is the minimum energy required to remove an electron from the metal surface. The relationship between the cut-off wavelength (λ_cutoff) and the work function (Φ) is given by λ_cutoff = hc / Φ, where h is Planck's constant (approximately 6.626 × 10^-34 J·s) and c is the speed of light (approximately 3.00 × 10^8 m/s). By substituting the value of the work function (4.25 eV) into the equation, we can calculate the cut-off wavelength of lead.
c) Once the wavelength of the emitted light from part a) is known, the velocity of the emitted electrons can be determined using the de Broglie wavelength equation: λ = h / mv, where m is the mass of the electron and v is its velocity. By rearranging the equation, we can solve for the velocity: v = h / (mλ). By substituting the mass of an electron and the calculated wavelength into the equation, we can find the velocity of the emitted electrons.
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If an electron (mass =9.1×10 −31
kg ) is released at a speed of 4.9×10 5
m/s in a direction perpendicular to a uniform magnetic field, then moves in a circle of radius 1.0 cm, what must be the magnitude of that field? μTx
The magnitude of the field is 1.41 × 10^-3 T.
When a charged particle moves in a magnetic field perpendicular to the magnetic field, the Lorentz force acts as a centripetal force causing the charged particle to move in a circle. The centripetal force is given by the relation: F = ma = (mv²)/r.
Where m is the mass of the charged particle, v is the velocity of the charged particle, r is the radius of the circle and a is the acceleration of the charged particle due to the magnetic field.Based on the information given in the question;Mass of the electron, m = 9.1 × 10^-31 kgVelocity of the electron, v = 4.9 × 10^5 m/s.
Radius of the circle, r = 1.0 cm = 0.01 mThe force acting on the electron due to the magnetic field is given by the relation: F = qvB. Where q is the charge of the electron, v is the velocity of the electron and B is the magnetic field strength.
Since the force acting on the electron is the centripetal force, equating these two forces we get: F = mv²/r = qvB. Therefore, B = mv/rq = (9.1 × 10^-31 kg × (4.9 × 10^5 m/s))/((0.01 m) × 1.6 × 10^-19 C) = 1.41 × 10^-3 T.So, the magnitude of the magnetic field is 1.41 × 10^-3 T.Answer: The magnitude of the field is 1.41 × 10^-3 T.
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A spherical UFO streaks across the sky at a speed of 0.90c relative to the earth. A person on earth determines the length of the UFO to be 230 m along the direction of its motion. State the ship's dimensions in the x- and y-axis as its travelling and when it lands (you must solve for the length/diameter of the ship).
The ship's dimensions in the x-axis are approximately 676.2 m (length) and D₀ (diameter), and its dimensions in the y-axis remain the same as D₀ when it is moving and when it lands.
To solve for the dimensions of the ship along the x- and y-axis, we can use the concept of length contraction in special relativity. According to special relativity, objects moving at high speeds relative to an observer undergo length contraction in the direction of their motion.
Let's denote the ship's dimensions in its rest frame (ship's frame) as L₀ (length) and D₀ (diameter). We want to find the dimensions of the ship as observed by a person on Earth when it is moving at a speed of 0.90c.
The length contraction factor, γ, can be calculated using the Lorentz factor:
γ = 1 / sqrt(1 - (v/c)^2)
Where v is the velocity of the ship and c is the speed of light.
Given that v = 0.90c, we can calculate γ:
γ = 1 / sqrt(1 - (0.90)^2)
Using a calculator, we find γ ≈ 2.94.
Now, let's consider the length contraction along the direction of motion (x-axis):
L = L₀ / γ
Substituting the given length (L) as 230 m, we can solve for L₀:
230 m = L₀ / 2.94
Solving for L₀, we find L₀ ≈ 676.2 m.
Therefore, the ship's length in its frame is approximately 676.2 m.
Next, let's consider the diameter along the y-axis. According to length contraction, there is no contraction in directions perpendicular to the motion. Therefore, the diameter of the ship remains the same:
D = D₀
Since no length contraction occurs along the y-axis, the ship's diameter remains unchanged.
The ship's dimensions in the x-axis are approximately 676.2 m (length) and D₀.
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Calculate the amplitude of the motion. An object with mass 3.2 kg is executing simple harmonic motion, attached to a spring with spring constant 310 N/m. When the object is 0.019 m from its equilibrium position, it is moving with a speed of 0.55 m/s. Express your answer to two significant figures and include the appropriate units. Mi ) ?Calculate the maximum speed attained by the object. Express your answer to two significant figures and include the appropriate units.
The maximum speed attained by the object is approximately 0.19 m/s. To calculate the amplitude of the motion, we can use the formula:
A = [tex]x_{max[/tex]
where A is the amplitude and [tex]x_{max[/tex] is the maximum displacement from the equilibrium position.
Given that the object is 0.019 m from its equilibrium position, we can conclude that the amplitude is also 0.019 m.
So, the amplitude of the motion is 0.019 m.
To calculate the maximum speed attained by the object, we can use the equation:
[tex]v_{max[/tex] = ω * A
where [tex]v_{max[/tex] is the maximum speed, ω is the angular frequency, and A is the amplitude.
The angular frequency can be calculated using the formula:
ω = √(k / m)
where k is the spring constant and m is the mass.
Given that the spring constant is 310 N/m and the mass is 3.2 kg, we can calculate ω:
ω = √(310 N/m / 3.2 kg)
≈ √(96.875 N/kg)
≈ 9.84 rad/s
Now we can calculate the maximum speed:
[tex]v_{max[/tex] = 9.84 rad/s * 0.019 m
≈ 0.19 m/s
Therefore, the maximum speed attained by the object is approximately 0.19 m/s.
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a) The general form of Newton's Law of cooling is: T(t) = Ta +(T(0) – Tale-ke where T is the temperature at any time, t in minutes. Ta is the surrounding ambient temperature in °C and k is the cooling con- stant. Consider a cup of coffee at an initial temperature, T(0) of 80°C placed into the open air at 15°C. After 5 minutes the coffee cools to 65°C. Using these initial conditions: i) Calculate the cooling constant, k. ii) What will be the temperature of the coffee after exactly 13 minutes? iii) How long will it take for the coffee to reach 25°C?
i) The cooling constant (k) is approximately 0.6667.
ii) After exactly 13 minutes, the temperature of the coffee will be around 19.3°C.
iii) It will take approximately 43.7 minutes for the coffee to reach a temperature of 25°C.
i) To calculate the cooling constant (k):
k = (T(0) - Ta - T(t)) / (T(t) - Ta)
= (80 - 15 - 65) / (65 - 15)
= 0.6667
ii) To find the temperature of the coffee after exactly 13 minutes, we can substitute t = 13, T(0) = 80, Ta = 15, and k = 0.6667 into the Newton's Law of cooling equation:
T(13) = 15 + (80 - 15 - 15)e(-0.6667*13) ≈ 19.3°C
iii) To determine the time required for the coffee to reach 25°C:
t = ln((T(0) - Ta) / (T(0) - T)) / k
= ln((80 - 15) / (80 - 25)) / 0.6667
≈ 43.7 minutes
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Two identical balls of clay are positioned such that one piece is located 4.8 meters directly above the other, which is on the ground. The upper piece of clay is released from rest while the lower one is shot straight up from the ground at a speed of 6 m/s. When the clay balls collide, they stick together. Find the speed of the balls when they strike the ground together.
Please explain thoroughly, some solutions do not explain. Please
Given that: The height of the ball above the ground, h = 4.8 metersThe initial velocity of the lower ball, u = 6 m/sNow, the initial velocity of the upper ball = 0 m/s, because it is released from rest.
Both the balls have the same mass and collide inelastically, which means the total momentum of the system is conserved. Let v be the velocity of the combined mass of both the balls after the collision. Since the momentum of the system is conserved, we can write the equation as:mu + 0 = (mu + mv)vWhere,m is the mass of each ballu is the initial velocity of the lower ballv is the velocity of the combined mass of both the balls after the collision.
Therefore,v = u/2 = 6/2 = 3 m/sThis is the velocity with which the combined mass of both the balls moves upwards after the collision. Now we can find the time, T it takes to reach the maximum height using the formula:T = (2h/v)T = (2 × 4.8)/3 = 3.2 sUsing this time, we can find the velocity with which the combined mass of both the balls strikes the ground using the formula:v = gtwhere g = 9.8 m/s²v = 9.8 × 3.2 = 31.36 m/s
Therefore, the speed of the balls when they strike the ground together is 31.36 m/s or approximately 31 m/s (rounded to two decimal places).Hence, the correct answer is 31 m/s.
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You are spending the summer as an assistant learning how to navigate on a large ship carrying freight across Lake Erie. One day, you and your ship are to travel across the lake a distance of 200 km traveling due north from your origin port to your destination port. Just as you leave your origin port, the navigation electronics go down. The cap- tain continues sailing, claiming he can depend on his years of experience on the water as a guide. The engineers work on the navigation system while the ship continues to sail, and winds and waves push it off course. Eventually, enough of the navigation system comes back up to tell you your location. The system tells you that your current position is 50.0 km north of the origin port and 25.0 km east of the port. The captain is a little embarrassed that his ship is so far off course and barks an order to you to tell him immedi- ately what heading he should set from your current position to the destination port. Give him an appropriate heading angle.
You should advise the captain to set a heading angle of approximately 63.43 degrees from your current position towards the destination port.
To determine the heading angle from your current position to the destination port, you can use trigonometry. Given that your current position is 50.0 km north and 25.0 km east of the origin port, you can consider these values as the lengths of the legs of a right triangle.
The desired heading angle can be found using the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the northward distance (50.0 km) and the adjacent side is the eastward distance (25.0 km).
The heading angle (θ) can be calculated as:
θ = tan^(-1)(opposite/adjacent)
θ = tan^(-1)(50.0 km/25.0 km)
Using a calculator, the approximate value of the heading angle is:
θ ≈ 63.43 degrees
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‒‒‒‒‒‒‒‒‒‒ A man pulls a 77 N sled at constant speed along a horizontal snow surface. He applies a force of 80 N at an angle of 53° above the surface. What is the normal force exerted on the sled? Q141N 77 N 64 N 13 N
The normal force exerted on the sled is 77N. The normal force is the force exerted by a surface perpendicular to the object resting on it.
In this scenario, the man is pulling the sled at a constant speed along a horizontal snow surface. The force he applies is 80 N at an angle of 53° above the surface. To determine the normal force exerted on the sled, we need to consider the forces acting on it.
The normal force is the force exerted by a surface perpendicular to the object resting on it. In this case, since the sled is on a horizontal surface, the normal force is directed vertically upwards to counteract the force of gravity. Since the sled is not accelerating vertically, the normal force is equal in magnitude but opposite in direction to the gravitational force acting on it.
The weight of the sled can be calculated using the equation F = mg, where m is the mass of the sled and g is the acceleration due to gravity (approximately [tex]9.8 m/s^2[/tex]). The weight of the sled is therefore 77 N. Since the sled is not accelerating vertically, the normal force exerted on it must be equal to its weight, which is 77 N.
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A toaster is rated at 660 W when connected to a 220 V source. What current does the toaster carry? A. 2.0 A B. 2.5 A C. 3.0 A D. 3.5 A
The given toaster is rated at 660 W when it is connected to a 220 V source. We can find the current that the toaster as follows,
P = VI or I=P/V, where P is the power, V is the voltage, I is the current
So, I=660/220
I=3A
Therefore, the current that the toaster carries C. 3.0 A.
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A circuit connects battery to three light bulbs in parallel. In other words, all the light bulbs are in parallel with one another, and in parallel with the battery. What happens to the circuit if one of the light bulb burns out? Why? A. Total resistance increases, other bulbs get brighter B. Total resistance increases, other bulbs get dimmer C. Total resistance increases, brightness of other bulbs does not change D. All the bulbs go out E. Total resistance decreases, other bulbs get brighter F. Total resistance decreases, other bulbs get dimmer G. Total resistance decreases, brightness of other bulbs does not change
If one of the light bulb burns out, Total resistance increases, other bulbs get dimmer. The circuit would not be broken if one of the bulbs burns out. This is the effect of a parallel circuit when one component fails. Therefore. the correct answer is option B.
In a parallel circuit, each device operates independently. As a result, if one component fails, it does not cause the others to stop working. However, since the resistance of each bulb is fixed, the total resistance of the circuit decreases as bulbs are added.
When a bulb burns out, the resistance of the circuit rises, making the other bulbs dimmer. Because the current in a parallel circuit is divided among the components, the current flowing through each remaining bulb would decrease if one bulb burns out.
So, if one bulb fails, the voltage across it would drop, and it would get dimmer. That's why in parallel circuit the bulbs are installed in parallel to ensure that they function independently of each other. So, option B is the correct answer.
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A source emits monochromatic light of wavelength 558 nm in air. When the light passes through a liquid, its wavelength reduces to 420 nm. (a) What is the liquid's index of refraction? (b) Find the speed of light in the liquid. m/s
Dividing the wavelength in air (558 nm) by the wavelength in the liquid (420 nm) will give the refractive index. The liquid's index of refraction is 1.33. The speed of light in liquid is [tex]2.26 x 10^8 m/s.[/tex]
(a) To calculate the refractive index of the liquid, we can use the formula: n = λ_air / λ_liquid
Substituting the given values of λ_air = 558 nm and λ_liquid = 420 nm into the formula, we have:
n = [tex]\frac{558}{420}[/tex]
Calculating the value:
n = 1.33
Therefore, the index of refraction of the liquid is approximately 1.33.
(b) To find the speed of light in the liquid, we can use the equation:
v = c / n
where v is the speed of light in the medium, c is the speed of light in a vacuum, and n is the index of refraction of the medium.
v = [tex]\frac{(3.0 x 10^8 m/s)}{1.33}[/tex]
Calculating the value:
v ≈ [tex]2.26 x 10^8 m/s[/tex]
Therefore, the speed of light in the liquid is approximately [tex]2.26 x 10^8 m/s.[/tex]
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A 17-cm-diameter circular loop of wire is placed in a 0.86-T magnetic field When the plane of the loop is perpendicular to the field ines, what is the magnetic flux through the loop? Express your answer to two significant figures and include the appropriate units. H Фа Value Units Submit Request Answer Part B The plane of the loop is rotated until it makes a 40 angle with the field lines. What is the angle in the equation 4 - BAcoso for this situation? Express your answer using two significant figures. Request Answer Part B A 17-cm-diameter circular loop of wire is placed in 0.86-T magnetic field The plane of the loop is rotated until it makes a 40"angle with the field lines. What is the angle in the equation = BA cos for this situation? Express your answer using two significant figures.
When plane circular loop wire is perpendicular magnetic field, magnetic flux through loop can be calculated using Φ = B * A. The angle in eq Φ = B * A * cos(θ) represents angle between the magnetic field lines and normal to loop.
In the first scenario where the plane of the loop is perpendicular to the magnetic field lines, we can calculate the magnetic flux through the loop using the formula Φ = B * A. The diameter of the loop is 17 cm, which corresponds to a radius of 8.5 cm or 0.085 m. The area of the loop can be calculated as A = π * r^2, where r is the radius. Substituting the values, we get A = π * (0.085 m)^2. The given magnetic field is 0.86 T. Plugging in the values, the magnetic flux Φ is equal to (0.86 T) multiplied by the area of the loop.
In the second scenario, the plane of the loop is rotated until it makes a 40° angle with the magnetic field lines. In the equation Φ = B * A * cos(θ), θ represents the angle between the magnetic field lines and the normal to the loop. Therefore, the given angle of 40° can be substituted into the equation to determine the contribution of the angle to the magnetic flux.
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Task 2
Activation Polarization is a mechanism that explains the
corrosion rate. Explain which part of the reaction determines the
total reaction rate.
Activation polarization is a mechanism that influences the corrosion rate, and it is the activation energy of the electrochemical reaction that determines the total reaction rate.
Activation polarization refers to the increase in the electrochemical reaction rate caused by the energy barrier, known as activation energy, that needs to be overcome for the reaction to proceed. The total reaction rate in corrosion is determined by the activation energy, which represents the minimum energy required for the reaction to occur.
In the context of corrosion, activation polarization occurs at the electrode-electrolyte interface. It is caused by various factors such as the nature of the corroding material, composition of the electrolyte, temperature, and presence of inhibitors. Activation polarization affects the rate of electrochemical reactions involved in the corrosion process.
When the activation energy is high, the reaction rate is low, leading to slower corrosion. On the other hand, when the activation energy is low, the reaction rate is high, resulting in faster corrosion. Therefore, the activation energy, which determines the activation polarization, plays a critical role in determining the total reaction rate of corrosion.
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8. [-12 Points] DETAILS SERCP11 22.7.P.037. A plastic light pipe has an index of refraction of 1.66. For total internal reflection, what is the mi (a) air 0 (b) water O Need Help? Read It MY NOTES ASK YOUR TEACHER internal reflection, what is the minimum angle of incidence if the pipe is in the following media? V MY NOTES ASK YOUR TEACHER
A plastic light pipe has an index of refraction of 1.66. for both (a) air and (b) water as the initial medium, total internal reflection does not occur when light enters the plastic light pipe with a refractive index of 1.66.
To determine the critical angle for total internal reflection, we can use Snell's law, which relates the angles of incidence and refraction at the interface between two media:
n1 × sin(theta1) = n2 × sin(theta2)
where:
n1 is the refractive index of the first medium (initial medium),
theta1 is the angle of incidence,
n2 is the refractive index of the second medium (final medium), and
theta2 is the angle of refraction.
For total internal reflection, the angle of refraction (theta2) becomes 90 degrees. Therefore, we can rewrite Snell's law as:
n1 × sin(theta1) = n2 × sin(90)
Since sin(90) = 1, the equation simplifies to:
n1 × sin(theta1) = n2
(a) Air as the initial medium:
Given n1 = 1 (approximating the refractive index of air as 1) and n2 = 1.66 (refractive index of the plastic light pipe), we can rearrange the equation to solve for sin(theta1):
sin(theta1) = n2 / n1
sin(theta1) = 1.66 / 1
sin(theta1) = 1.66
However, the sine of an angle cannot be greater than 1. Therefore, there is no critical angle for total internal reflection when light travels from air to the plastic light pipe. Total internal reflection does not occur in this case.
(b) Water as the initial medium:
Given n1 = 1.33 (refractive index of water) and n2 = 1.66 (refractive index of the plastic light pipe), we can use the same equation to find sin(theta1):
sin(theta1) = n2 / n1
sin(theta1) = 1.66 / 1.33
sin(theta1) ≈ 1.248
To find the angle theta1, we can take the inverse sine of sin(theta1):
theta1 = arcsin(sin(theta1))
theta1 ≈ arcsin(1.248)
However, since the sine of an angle cannot exceed 1, there is no real solution for theta1 in this case. Total internal reflection does not occur when light travels from water to the plastic light pipe.
Therefore, for both (a) air and (b) water as the initial medium, total internal reflection does not occur when light enters the plastic light pipe with a refractive index of 1.66.
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An electron (mass 9 x 10⁻³¹ kg) is traveling at a speed of 0.91c in an electron accelerator. An electric force of 1.6 x 10 N is applied in the direction of motion while the electron travels a distance of 2 m. You need to find the new speed of the electron. Which of the following steps must be included in your solution to this problem? (a) Calculate the initial particle energy Yimc of the electron. (b) Calculate the final particle energy y&mc? of the electron. (c) Determine how much time it takes to move this distance. (d) Use the expression m[512 to find the kinetic energy of the electron. (e) Calculate the net work done on the electron. (f) Use the final energy of the electron to find its final speed. What is the new speed of the electron as a fraction of c?
The new speed of the electron as a fraction of c is 0.9655.
Mass of electron = m = 9 x 10⁻³¹ kg
Speed of electron = u = 0.91c
Electric force = F = 1.6 x 10 N
Crossing distance = s = 2 m
Electric force = F = ma
where, F = Electric force, m = Mass of the electron, a = Acceleration of the electron.
Using above equation, we get, a = F/ma = F/m = 1.6 x 10 / 9 x 10⁻³¹ a = 1.78 x 10⁴ m/s²
Now, we can calculate the time taken by electron to travel a distance of 2m using s = ut + ½ at²
where, u = Initial speed of electron, t = Time taken by electron to travel distance s, a = Acceleration of electron, s = Distance travelled by electron.
So, t = s / (u/2 + ½ a)
We get, t = 2 / [0.91c/2 + 1/2 * 1.78 x 10⁴]
= 5.71 x 10⁻¹⁰ s
Kinetic energy = [m / √(1- (v/c)²)] c² - mc²
where, Kinetic energy = Final kinetic energy of electron, m = Mass of the electron, v = Final speed of the electron.
So, K.E = [9 x 10⁻³¹ / √(1-(v/c)²)] c² - (9 x 10⁻³¹) c²
Now, calculate the net work done on the electron. Wnet = K.E - K.Eo
where, Wnet = Net work done on electron, K.E = Final kinetic energy of electron, K.Eo = Initial kinetic energy of electron.
K.Eo = [9 x 10⁻³¹ / √(1-(u/c)²)] c² - (9 x 10⁻³¹) c²
we get, Wnet = [9 x 10⁻³¹ / √(1-(v/c)²)] c² - [9 x 10⁻³¹ / √(1-(u/c)²)] c²
Simplify this expression, Wnet = 0.5 x 9 x 10⁻³¹ [(1/√(1-(v/c)²)] c² - [(1/√(1-(u/c)²)] c²
= 0.5 x m [(1/√(1-(v/c)²)] c² - [(1/√(1-(u/c)²)] c²
Finally, use the work-energy principle. We know that, Wnet = ΔK.E
Wnet = Work done on the particle, ΔK.E = Change in kinetic energy of the particle.
Since the electron is being accelerated, the force acting on it is in the same direction as the direction of motion and hence, the work done is positive. So, we can write Wnet = K.E - K.Eo.
Now, put the values of Wnet, ΔK.E, K.E and K.Eo, we get,0.5 x m [(1/√(1-(v/c)²)] c² - [(1/√(1-(u/c)²)] c²
= [(9 x 10⁻³¹ / √(1-(v/c)²)] c² - [(9 x 10⁻³¹ / √(1-(u/c)²)] c² - [(9 x 10⁻³¹ / √(1-(u/c)²)] c²
Now, we can calculate the final kinetic energy of the electron, Kinetic energy = (Wnet + K.Eo)K.E = 0.5 x m [(1/√(1-(v/c)²)] c² + [(9 x 10⁻³¹ / √(1-(u/c)²)] c²K.E
= [9 x 10⁻³¹ / √(1-(v/c)²)] c²v/c
= √[1 - ((m/m+1)(c/u²t²))]v/c
= √[1 - ((9 x 10⁻³¹/10⁻³¹ + 1)(3 x 10⁸/(0.91 x 3 x 10⁸)² x (5.71 x 10⁻¹⁰)²))]v/c = 0.9655
Therefore, the new speed of the electron as a fraction of c is 0.9655.
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A parallel-plate capacitor has plates of dimensions 2.0 cm by 3.0 cm separated by a 1.0- olaviomm thickness of dielectric material (k = 11.1), what is its capacitance? C. 60 pF D. 80 pF A. 20 pF B. 40 pF 5. A spherical liquid drop of radius R has a capacitance of C = 4πER. If two such drops combine to form a single larger drop, what is its capacitance? A A. 2 C B. C C. 1.26 C D. 1.46 C
The capacitance of the parallel-plate capacitor is approximately 5.31 x 10⁻¹¹ F or 53.1 pF. To find the capacitance of a parallel-plate capacitor, we can use the formula:
C = (ε₀ * εᵣ * A) / d
where:
C is the capacitance,
ε₀ is the vacuum permittivity (8.854 x 10⁻¹² F/m),
εᵣ is the relative permittivity or dielectric constant (given as 11.1),
A is the area of the plates (2.0 cm by 3.0 cm = 0.02 m * 0.03 m = 0.0006 m²),
d is the separation between the plates (1.0 mm = 0.001 m).
Plugging in the values, we have:
C = (8.854 x 10⁻¹² F/m * 11.1 * 0.0006 m²) / 0.001 m
= 5.31 x 10⁻¹¹ F
Therefore, the capacitance of the parallel-plate capacitor is approximately 5.31 x 10⁻¹¹ F or 53.1 pF.
For the second part of the question, when two identical drops combine to form a larger drop, the total capacitance is given by the sum of the individual capacitances:
C_total = C1 + C2
Since each individual drop has a capacitance of C, we have:
C_total = C + C = 2C
Therefore, the capacitance of the single larger drop formed by combining two identical drops is 2 times the original capacitance, which is 2C. In this case, it is given that C = 4πER, so the capacitance of the single larger drop is 2 times that:
C_total = 2C = 2(4πER) = 8πER
Hence, the capacitance of the single larger drop is 8πER.
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When this astronaut goes
back to Earth, what will
happen?
A. His weight will increase.
B. His mass will increase.
C. Both his mass and weight will decrease.
Answer: A
Explanation: The mass of a thing never changes but weight is the act of gravity on mass. This rules out B and C since mass can’t change. Leaving A as the only possible answer.
Sketch and label the equivalent circuit of DC series motor and DC compound generator b) A 220 V DC series motor runs at 800 rpm and takes 30A. The value of the armature and field resistance are 0.6 ≤ and 0.8 №, respectively. Determine: i. The back EMF. a) ii. iii. The torque developed in the armature. The output power if rotational losses are 250 W.
In the case of the DC series motor, the back EMF of the motor is 202 V.
The equivalent circuit of a DC series motor and DC compound generator can be represented as follows:
The armature resistance (Ra) is connected in series with the armature winding.
The field resistance (Rf) is connected in series with the field winding.
The back electromotive force (EMF) (Eb) opposes the applied voltage (V).
For the specific case mentioned:
Given:
Applied voltage (V) = 220 V
Speed (N) = 800 rpm
Current (I) = 30 A
Armature resistance (Ra) = 0.6 Ω
Field resistance (Rf) = 0.8 Ω
To calculate the back EMF (Eb) of the motor, we can use the following formula:
Eb = V - I * Ra
Substituting the given values:
Eb = 220 V - 30 A * 0.6 Ω
= 220 V - 18 V
= 202 V
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--The complete Question is, What is the equivalent circuit of a DC series motor and DC compound generator? In a specific case, a 220 V DC series motor runs at 800 rpm and draws a current of 30A. The armature resistance is 0.6 Ω, and the field resistance is 0.8 Ω. Calculate the back EMF of the motor.--
An LED (Light Emitting Diode) is constructed from a p-n junction based on a certain semi-conducting material with a band gap of 1.61 eV. What is the wavelength of the emitted light? Give your answer to the closest nm (no decimal places). Do not include the units.
The light-emitting diode (LED) is a two-terminal semiconductor light source used as a light source in lighting. The wavelength of the emitted light from the LED is 1240.
An LED (light-emitting diode) is made up of a p-n junction made of a particular semiconducting substance with a bandgap of 1.61 eV. The wavelength of the emitted light is given in this question and needs to be calculated.
The energy of the photon is related to the wavelength λ by the formula,
E = hc/λ
where E is the photon energy, h is Planck's constant, and c is the speed of light.
The formula can be modified to find the wavelength of the emitted light:
λ = hc/E
where λ is the wavelength, h is Planck's constant, c is the speed of light, and E is the energy of a photon.
The energy gap of the p-n junction of an LED determines the energy and frequency of the photon emitted.
The energy gap is given in the question to be 1.61 eV.
h and c are constants that are well-known.
The value of h is 6.626 x 10-34 joule-second, and c is 2.998 x 108 meter/second.
Substituting the values,
λ = hc/Eλ
= (6.626 x 10-34) x (2.998 x 108) / (1.61 x 1.6 x 10-19)λ
= 1.24 x 10-6 meter
= 1240 nm
Therefore, the wavelength of the emitted light from the LED is 1240 nm.
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If 200 m away from an ambulance siren the sound intensity level is 65 dB, what is the sound intensity level 20 m away from that ambulance siren? Specify your answer in units of decibel (dB). \begin{tabular}{|llllll} \hline A: 75 & B: 80 & C: 85 & D: 90 & E: 95
The sound intensity level at 20 m from the ambulance siren is:x = 2.5 dB
The given problem states that the sound intensity level at a distance of 200 m from an ambulance siren is 65 dB and we need to calculate the sound intensity level at 20 m from the siren. Let us assume that the sound intensity level at a distance of 20 m from the siren be x dB.
Now we know that the sound intensity level at any point is given by the following formula: IL = 10log(I/I0), where I is the sound intensity and I0 is the threshold of hearing, which is equal to 10^-12 W/m^2.
So the sound intensity level 200 m away from the ambulance siren, which is 65 dB, can be written as:
65 = 10log(I/10^-12)
65/10 = log(I/10^-12)
6.5 = log(I/10^-12)I/10^-12 = antilog(6.5)I/10^-12 = 3.162 * 10^-7 W/m^2
Similarly, the sound intensity level at a distance of 20 m from the ambulance siren, which is x dB, can be written as:x = 10log(I/10^-12)x/10 = log(I/10^-12)x/10 = log(I) - log(10^-12)x/10 = log(I) + 12/10x/10 - 12 = log(I)I/10^-12 = antilog(x/10 - 12)I/10^-12 = 10^(x/10) * 10^-12 W/m^2
Since the sound intensity level remains constant, the sound intensity at a distance of 200 m and 20 m is the same. Therefore, equating the above two expressions, we get:3.162 * 10^-7 = 10^(x/10) * 10^-12 3.162 = 10^(x/10)10^(x/10) = 3.162
Taking the logarithm of both sides, we get:x/10 = log(3.162)x/10 = 0.5x = 5log(3.162)x = 5 * 0.5x = 2.5
Therefore, the sound intensity level at 20 m from the ambulance siren is:x = 2.5 dB
Sound intensity level at 20 m from the ambulance siren is 2.5 dB.
Answer: 2.5 dB
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The correct answer is: A,Aω,Aω2 The position of an object moving in simple harmonic motion is given by the equation x(t)=Asin(ωt+θ), where A=−3.7 m, at=2.0rad/s and θ=0.20rad. What is the speed of the object when it is at x=−1.5 m ? Select one: a. 7.0 m/s b. 6.8 m/s c. 3.8 m/s d. 3.4 m/s Take the denvative of x(t) to find the velocity as a function of tate: x(t)=Asin(ωt+θ)v(t)=dtdx
The speed of the object when it is at x = -1.5 m is 7.0 m/s. Answer: a. 7.0 m/s.
Given data,A = -3.7 mω = 2.0 rad/st = ?θ = 0.20 radWe know that velocity as a function of time is given by the derivative of position as a function of time, that is,v(t) = d/dt [x(t)]v(t) = d/dt [Asin(ωt + θ)]v(t) = Aω cos(ωt + θ)Now, the position of the object is given byx(t) = Asin(ωt + θ)Now, substituting the given values, we getx(t) = -3.7 sin(2t + 0.20) mNow, the object is at x = -1.5 mHence, -1.5 = -3.7 sin(2t + 0.20)Solving for t, we gett = 0.835 sNow, substituting t = 0.835 s in the equation of velocity as a function of time, we getv(t) = Aω cos(ωt + θ)v(t) = -3.7 × 2.0 cos(2(0.835) + 0.20) m/sv(t) = -7.0 m/sTherefore, the speed of the object when it is at x = -1.5 m is 7.0 m/s. Answer: a. 7.0 m/s.
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It was found that an EM wave is comprised of individual spherical particles. These spherical paticles form the resulting wowe-foont This coss Critical angle Snell's Law Wave cavity Brewster's Angle Coulomb's Law wavegulde Huygens ndividual sphencal particles. These spherical particles form the resulting wave-front. This observation is known as...
The phenomenon of EM waves composed of individual spherical particles that form the resulting wavefront is referred to as Huygens Principle.
Christiaan Huygens was a Dutch scientist who suggested in 1678 that every point on the primary wavefront acts as a source of secondary waves. These secondary waves are spherical waves that propagate at the same speed and frequency as the primary wave, but with different amplitudes and phases.Huygens principle aids in determining how waves behave when they interact with obstacles. It allows us to predict how a wave will propagate through a given geometry by imagining it as the sum of secondary wavelets produced by the primary wave.
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Water flows through a garden hose (radius =1.5 cm ) and fills a tub of volume V=200 Liters in Δt=5.6 minutes. What is the speed of the water in the hose in meters per second? Your Answer: Answer Question 15 (6 points) A beach ball is filled with air and has a radius of r=49 cm. How much mass would be needed to pull the ball underwater in a swimming pool? Answer in kg and assume the volume of the added weight is negligible.
Water flows through a garden hose and fills a tub of 200 Liters in 5.6 minutes. The speed of the water in the hose 0.841 meters per second. A beach ball is filled with air and has a radius of 49 cm and around 513.3 kg of mass is needed to pull the beach ball underwater in a swimming pool.
(a) To calculate the speed of water in the hose, we need to determine the flow rate. First, let's convert the volume of water from liters to cubic meters. Since 1 liter is equal to 0.001 cubic meters, we have:
Volume = 200 liters * 0.001 cubic meters/liter = 0.2 cubic meters
Next, let's convert the time from minutes to seconds:
Time = 5.6 minutes * 60 seconds/minute = 336 seconds
The flow rate (Q) can be calculated by dividing the volume by the time:
Q = [tex]\frac{Volume}{Time} }{}[/tex] = [tex]\frac{ 0.2 }{336}[/tex] = 0.0005952 cubic meters per second
The cross-sectional area of a circular hose can be calculated using the formula: Area =[tex]π * radius^2[/tex]
Given a radius of 1.5 cm, which is 0.015 meters, we have:
Area = [tex]π * (0.015 meters)^2[/tex] ≈ 0.00070686 square meters
Now we can calculate the speed (v) using the formula:
v = Q / Area = [tex]\frac{0.0005952}{0.00070686}[/tex] square meters ≈ 0.841 meters per second
Therefore, the speed of the water in the hose is approximately 0.841 meters per second.
(b) The volume of a sphere can be calculated using the formula:
Volume = [tex](\frac{4}{3} ) * π * radius^3[/tex]
Given a radius of 49 cm, which is 0.49 meters, we have:
Volume = [tex](\frac{4}{3} ) * π * 0.49^3[/tex] ≈ 0.512 cubic meters
The density of water is approximately 1000 kg/m^3. Therefore, the weight of the water displaced by the ball is:
Weight of water displaced = Volume * Density * gravitational acceleration
= 0.512 cubic meters * [tex]1000 kg/m^3 * 9.8 m/s^2[/tex]
≈ 5025.6 Newtons
To balance the buoyant force, an equal and opposite gravitational force is required. The gravitational force is given by:
Gravitational force = Mass * gravitational acceleration
To find the mass needed to balance the buoyant force, we divide the weight of water displaced by the gravitational acceleration:
Mass = Weight of water displaced / gravitational acceleration
=[tex]\frac{5025.6 Newtons}{9.8 m/s^2}[/tex]
≈ 513.3 kg
Therefore, approximately 513.3 kg of mass would be needed to pull the beach ball underwater in a swimming pool.
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a) At what frequency would a 6.0 mH inductor and a 10 nF capacitor have the same reactance? (b) What would the reactance be? (©) Show that this frequency would be the nat- ural frequency of an oscillating circuit with the same L and C.
Answer:
The frequency at which the 6.0 mH inductor and 10 nF capacitor have the same reactance is approximately 20,462 Hz.
Reactance of an inductor (XL) is given by:
XL = 2πfL
Reactance of a capacitor (XC) is given by:
XC = 1 / (2πfC)
Where f is the frequency, L is the inductance, and C is the capacitance.
Setting XL equal to XC:
2πfL = 1 / (2πfC)
Simplifying the equation:
f = 1 / (2π√(LC))
L = 6.0 mH
= 6.0 x 10^(-3) H
C = 10 nF
= 10 x 10^(-9) F
Substituting the given values into the equation:
f = 1 / (2π√(6.0 x 10^(-3) H * 10 x 10^(-9) F))
Simplifying the expression:
f = 1 / (2π√(60 x 10^(-12) H·F))
f = 1 / (2π√(60 x 10^(-12) s^2 / C^2))
f = 1 / (2π x 7.75 x 10^(-6) s)
f ≈ 20,462 Hz
Therefore, the frequency at which the 6.0 mH inductor and 10 nF capacitor have the same reactance is approximately 20,462 Hz.
To show that this frequency is the natural frequency of an oscillating circuit with the same L and C, we can use the formula for the natural frequency of an LC circuit:
fn = 1 / (2π√(LC))
Substituting the given values into the formula:
fn = 1 / (2π√(6.0 x 10^(-3) H * 10 x 10^(-9) F))
fn = 1 / (2π√(60 x 10^(-12) H·F))
fn = 1 / (2π√(60 x 10^(-12) s^2 / C^2))
fn = 1 / (2π x 7.75 x 10^(-6) s)
fn ≈ 20,462 Hz
We can see that this frequency matches the frequency obtained earlier, confirming that it is the natural frequency of an oscillating circuit with the same L and C.
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