When light with a frequency of 5.4×10^14 Hz enters a liquid with a refractive index of 2.0, its frequency will remain the same.
The frequency of light refers to the number of complete oscillations or cycles it undergoes per unit of time. The index of refraction, denoted by "n," is a property of a medium that describes how light propagates through it.
It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. In this case, the light enters a liquid with a refractive index of 2.0.
When light passes from one medium to another, its speed and wavelength change, while the frequency remains constant. The frequency of light is determined by the source and remains constant regardless of the medium it traverses.
Therefore, the frequency of light with a value of 5.4×10^14 Hz will remain the same when it enters the liquid with a refractive index of 2.0.In summary, the frequency of light with a vacuum frequency of 5.4×10^14 Hz will not change when it enters a liquid with a refractive index of 2.0.
The index of refraction only affects the speed and wavelength of light, while the frequency remains constant throughout different media.
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An object is placed 10cm in front of a concave mirror whose radius of curvature is 10cm calculate the position ,nature and magnification of the image produced
Answer:
The focal length, f = − 15 2 c m = − 7.5 c m The object distance, u = -10 cm Now from the mirror equation 1 v + 1 u = 1 f 1 v + 1 − 10 = 1 − 7.5 v = 10 × 7.5 − 2.5 = − 30 c m The image is 30 cm from the mirror on the same side as the object.
P.(s) may be converted to PH3(g) with H₂(g). The standard Gibbs energy of formation of PH3(g) is +13.4 kJ mol at 298 K. What is the corresponding reaction Gibbs energy when the partial pressures of the H2 and PH3 (treated as perfect gases) are 1.0 bar and 0.60 bar, respectively? What is the spontaneous direction of the reaction in this case?
The reaction Gibbs energy when the partial pressures of H2 and PH3 are 1.0 bar and 0.6 bar, respectively, is +12.1 kJ/mol. In this case, the reverse reaction is spontaneous.
The reaction Gibbs energy (ΔG_rxn) can be calculated using the equation:
ΔG_rxn = ΣnΔGf(products) - ΣnΔGf(reactants)
Given that the standard Gibbs energy of formation (ΔGf) of PH3(g) is +13.4 kJ/mol, we can substitute this value into the equation:
ΔG_rxn = (1 mol × 0 kJ/mol) - (1 mol × (+13.4 kJ/mol))
Simplifying the equation, we get:
ΔG_rxn = -13.4 kJ/mol
Since the reaction Gibbs energy is negative, the forward reaction is not spontaneous. However, the reverse reaction is spontaneous, indicated by the positive value of the reaction Gibbs energy. This means that the reaction will tend to proceed in the reverse direction, from PH3 to H2.
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Watching a car recede at 21 m/s, you notice that after 11 min the two taillights are no longer resolvable. If the diameter of your pupil is 5.0 mm in the dim ambient lighting, explain the reasoning for the steps that allow you to determine the spacing of the lights.
To determine the spacing of the taillights, you can use the concept of angular resolution. By considering the speed of the receding car, the time elapsed, the diameter of your pupil, and the distance traveled by the car, you can calculate the spacing between the taillights.
When observing a receding car, the spacing between its taillights can be determined by considering the concept of angular resolution. Angular resolution refers to the smallest angle at which two objects can be distinguished. In this scenario, you first convert the given time of 11 minutes to seconds (660 seconds) and calculate the distance traveled by the car during that time using its speed of 21 m/s (13,860 meters).
To determine the spacing between the taillights, you need to consider your line of sight. The diameter of your pupil, given as 5.0 mm, is converted to meters (0.005 meters). The angular resolution is then determined by dividing the diameter of your pupil by the distance between the taillights. By multiplying the angular resolution by the distance traveled by the car, you can calculate the spacing between the taillights. In this case, the spacing is equal to 0.005 meters.
Therefore, by following these steps and considering the relevant variables, you can determine the spacing between the taillights based on the concept of angular resolution and the given parameters.
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You are sitting in a bus in a depot, when suddenly you see in the window the bus next to yours start to move forward. List two scenarios that could be happening
An experimental bicycle wheel is place on a test stand so that it is free to turn on its axle. If a constant net torque of 7.5 N-m is applied to the tire for 1.5 seconds, the angular speed of the tire increases from 0 to 2 rev/min. The external torque is then removed, and the wheel is brought to rest by friction in its bearings in 175 s. a) Compute the moment of inertia of the wheel about the rotation rate. b) Compute the friction torque. c) Compute the total number of revolutions made by the wheel in the 175-second time interval.
Answer:
The total number of revolutions made by the wheel in the 175-second time interval is approximately 8.75 revolutions.
a) To compute the moment of inertia of the wheel about the rotation axis, we can use the equation:
Δθ = (1/2)αt^2
Where Δθ is the change in angle (in radians), α is the angular acceleration (in radians per second squared), and t is the time (in seconds).
Initial angular velocity, ω_i = 0 rev/min
Final angular velocity, ω_f = 2 rev/min
Time, t = 1.5 s
First, let's convert the angular velocities to radians per second:
ω_i = (0 rev/min) * (2π rad/rev) * (1 min/60 s) = 0 rad/s
ω_f = (2 rev/min) * (2π rad/rev) * (1 min/60 s) = (2π/30) rad/s
The angular acceleration can be calculated using the equation:
α = (ω_f - ω_i) / t
α = [(2π/30) rad/s - 0 rad/s] / 1.5 s = (2π/30) rad/s^2
Now, let's find the change in angle:
Δθ = (1/2) * (2π/30) rad/s^2 * (1.5 s)^2
Δθ = (π/30) rad
The moment of inertia (I) of the wheel can be determined using the equation:
Δθ = (1/2)αt^2 = (1/2) * (I * α) * t^2
Rearranging the equation:
I = (2Δθ) / (α * t^2)
Substituting the values:
I = (2 * π/30) rad / ((2π/30) rad/s^2 * (1.5 s)^2)
I = 2.222 kg·m^2
b) To compute the friction torque, we can use the equation:
τ_f = I * α
Substituting the values:
τ_f = (2.222 kg·m^2) * (2π/30) rad/s^2
τ_f ≈ 0.370 N·m
c) To compute the total number of revolutions made by the wheel in the 175-second time interval, we can use the equation:
Δθ = ω_avg * t
Where Δθ is the change in angle (in radians), ω_avg is the average angular velocity (in radians per second), and t is the time (in seconds).
Time, t = 175 s
First, let's calculate the average angular velocity:
ω_avg = (ω_i + ω_f) / 2 = (0 rad/s + (2π/30) rad/s) / 2 = (π/30) rad/s
Now, we can find the change in angle:
Δθ = (π/30) rad/s * 175 s
Δθ = 175π/30 rad ≈ 18.333π rad
To calculate the number of revolutions, we divide the change in angle by 2π:
Number of revolutions = (175π/30 rad) / (2π rad/rev) ≈ 8.75 rev
Therefore, the total number of revolutions made by the wheel in the 175-second time interval is approximately 8.75 revolutions.
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Help: The diagram below illustrates a light ray bouncing off a surface. Fill in the boxes with the correct terms.
The correct terms that fills the box are;
(i) The incident ray
(ii) The normal
(iii) The reflected ray
(iv) The angle of incident
(v) The reflected angle
What is the terms of the ray diagram?The terms of the ray diagram is illustrated as follows;
(i) This arrow indicates the incident ray, which is known as the incoming ray.
(ii) This arrow indicates the normal, a perpendicular line to the plane of incidence.
(iii) This arrow indicates the reflected ray; the out going arrow.
(iv) This the angle of incident or incident angle.
(v) This is the reflected angle or angle of reflection.
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Four point charges are held fixed in space on the corners of a rectangle with a length of 20 [cm] (in the horizontal direction) and a width of 10 [cm] (in the vertical direction). Starting with the top left corner and going clockwise, the charges are 9,=+10[nC], 92=-10[nC], 93=-5[nC), and 94=+8[nC). a) Find the magnitude and direction of the electric force on charge 94 b) Find the magnitude and direction of the electric field at the midpoint between 93 and 94 c) Find the magnitude and direction of the electric field at the center of the rectangle.
The magnitude of the electric force on charge 94 is approximately 4.81125 N. The direction can be determined based on the resultant vector of the individual forces.
To solve this problem, let's calculate the electric force and electric field step by step:
a) Magnitude and direction of the electric force on charge 94:
The electric force between two charges can be calculated using Coulomb's Law:
Electric force (F) = (k * |q1 * q2|) / r^2
where k is the electrostatic constant (k ≈ 8.99 × 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
We need to calculate the net force on charge 94 due to the other charges. Let's calculate the force individually for each pair of charges and then find the vector sum:
Force on charge 94 due to charge 91:
F_941 = (k * |q9 * q1|) / r_941^2
Force on charge 94 due to charge 92:
F_942 = (k * |q9 * q2|) / r_942^2
Force on charge 94 due to charge 93:
F_943 = (k * |q9 * q3|) / r_943^2
To find the net force, we need to consider the direction as well. Since the charges are held fixed, the net force should be in the direction of the resultant vector of the individual forces.
Net force on charge 94 = F_941 + F_942 + F_943
Calculate the distances between the charges:
r_941 = diagonal length of rectangle
r_942 = length of rectangle
r_943 = diagonal length of rectangle
Given:
Length of rectangle (L) = 20 cm = 0.2 m
Width of rectangle (W) = 10 cm = 0.1 m
Using the Pythagorean theorem:
Diagonal length of rectangle (d) = √(L^2 + W^2)
= √((0.2 m)^2 + (0.1 m)^2)
= √(0.04 m^2 + 0.01 m^2)
= √(0.05 m^2)
= 0.2236 m
Substituting the values, we can calculate the forces:
F_941 = (8.99 × 10^9 N m^2/C^2 * |8 × 10^(-9) C * 10 × 10^(-9) C|) / (0.2236 m)^2
≈ 1.815 N
F_942 = (8.99 × 10^9 N m^2/C^2 * |8 × 10^(-9) C * (-10) × 10^(-9) C|) / (0.2 m)^2
≈ 1.9975 N
F_943 = (8.99 × 10^9 N m^2/C^2 * |8 × 10^(-9) C * (-5) × 10^(-9) C|) / (0.2236 m)^2
≈ 0.99875 N
Now, calculate the net force:
Net force on charge 94 = F_941 + F_942 + F_943
= 1.815 N + 1.9975 N + 0.99875 N
≈ 4.81125 N
The magnitude of the electric force on charge 94 is approximately 4.81125 N. The direction can be determined based on the resultant vector of the individual forces.
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A glass sheet 1.30 μm thick is suspended in air. In reflected light, there are gaps in the visible spectrum at 547 nm and 615.00 nm. Calculate the minimum value of the index of refraction of the glass sheet that produces this effect.
The case of light reflected from the upper surface of the film, we found that the minimum value of the refractive index of the glass sheet that produces the gaps in the visible spectrum at 547 nm and 615.00 nm is 1.466. Therefore, we can conclude that this is the answer.
Given data:Thickness of glass sheet (t) = 1.30 μmGaps in the visible spectrum at 547 nm and 615.00 nmWe know that when light is reflected from a thin film, we see colored fringes due to interference of light waves.
The conditions for minimum reflection from a thin film are:When the thickness of the film is odd multiples of λ/4 i.e. t = (2n+1)(λ/4)when there is no phase change at the reflection i.e. when the reflected wave is in phase with the incoming wave.
Assuming the light is reflecting from the upper surface of the film, we can find the refractive index (n) of the glass sheet using the formula: t = [(2n + 1) λ1]/4where λ1 is the wavelength of light in air.The gaps are seen at λ = 547 nm and λ = 615 nm
Therefore, applying above formulae for both wavelengths and taking the difference of the refractive indices: t = [(2n + 1) λ1]/4When λ = 547 nm ⇒ λ1 = λ/n = 547/nTherefore, t = [(2n + 1) λ]/4⇒ 1.3 × 10⁻⁶ = [(2n + 1) × 547 × 10⁻⁹]/4⇒ 2n + 1 = 4 × 1.3/547 ⇒ 2n + 1 = 0.0095n = 2⇒ Refractive index (n) = λ/λ1 = 547/λ1t = [(2n + 1) λ1]/4When λ = 615 nm ⇒ λ1 = λ/n = 615/n
Therefore, t = [(2n + 1) λ]/4⇒ 1.3 × 10⁻⁶ = [(2n + 1) × 615 × 10⁻⁹]/4⇒ 2n + 1 = 4 × 1.3/615 ⇒ 2n + 1 = 0.0085n = 2⇒ Refractive index (n) = λ/λ1 = 615/nDifference in refractive indices (Δn) = n(λ=547) - n(λ=615)= 547/n - 615/n = 547/2 - 615/2= -34To produce the effect of minimum reflection, the minimum value of the refractive index of the glass sheet is 1.5 - 0.034 = 1.466.
For the case of light reflected from the upper surface of the film, we found that the minimum value of the refractive index of the glass sheet that produces the gaps in the visible spectrum at 547 nm and 615.00 nm is 1.466. Therefore, we can conclude that this is the answer.
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I will give brainliest to whoever answers all three asap :)
1. A 1.0 g insect flying at 2.0 km/h collides head-on with an 800 kg, compact car travelling at 90 km/h. Which object experiences the greater change in momentum during the collision?
a) Neither object experiences a change in momentum
b) The insect experiences the greater change in momentum
c) The compact car experiences the greater change in momentum
d) Both objects experience the same, non-zero change in momentum
2. Why are hockey and football helmets well padded?
a) to decrease the time of a collision, decreasing the force to the head
b) to decrease the time of a collision, increasing the force to the head
c) to increase the time of a collision, decreasing the force to the head
d) to increase the time of a collision, Increasing the force to the head
3. A 68.5 kg man and a 41.0 kg woman are standing at rest before performing a figure skating routine. At the start of the routine, the two skaters push off against each other, giving the woman a velocity of 3.25 m/s [N]. Assuming there is no friction between the skate blades and the ice, what is the man's velocity due to their push?
1. b) The insect experiences the greater change in momentum during the collision.
2. C) Hockey and football helmets are well padded to increase the time of a collision, decreasing the force to the head
3. The man's velocity due to their push is 0 m/s.
1. B. The insect experiences the greater change in momentum during the collision. Change in momentum is given by the formula Δp = mΔv, where Δp is the change in momentum, m is the mass, and Δv is the change in velocity. Although the mass of the car is much larger than the insect, the change in velocity experienced by the insect is significantly greater. Since the insect collides head-on with the car, its velocity changes from 2.0 km/h to nearly zero, resulting in a substantial change in momentum. On the other hand, the change in velocity of the car is relatively small since it collides with an object of much smaller mass. Therefore, the insect experiences the greater change in momentum.
2. Hockey and football helmets are well padded C. to increase the time of a collision, decreasing the force to the head. The padding in the helmets acts as a cushion, which extends the duration of the collision between the helmet and an object, such as a puck or a player. By increasing the collision time, the force experienced by the head is reduced. This is because the force of impact is given by the equation F = Δp/Δt, where F is the force, Δp is the change in momentum, and Δt is the change in time. By increasing the time, the force is spread out over a longer duration, resulting in a decrease in the force exerted on the head.
3. To determine the man's velocity due to their push, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the push is equal to the total momentum after the push. Since the woman has a velocity of 3.25 m/s [N] after the push, the man's velocity can be calculated as follows:
Total initial momentum = Total final momentum
(0 kg) + (41.0 kg)(0 m/s) = (68.5 kg + 41.0 kg)(v)
Simplifying the equation, we find:
0 = 109.5 kg * v
Dividing both sides by 109.5 kg, we get:
v = 0 m/s
Therefore, the man's velocity due to their push is 0 m/s. This means that he remains at rest while the woman gains velocity in the north direction.
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A clock has a 10.0-g mass object bouncing on a spring that has a force constant of 0.9 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? Umax m/s How many joules of kinetic energy does the object have at its maximum velocity? KEmax x 10-4 -
A clock has a 10.0-g mass object bouncing on a spring that has a force constant of 0.9 N/m. the object has approximately 1.08 x 10^(-3) J of kinetic energy at its maximum velocity.
To find the maximum velocity of the object bouncing on the spring, we can use the principle of conservation of mechanical energy.
The maximum potential energy of the object can be calculated when it reaches its maximum displacement from the equilibrium position. Since the object bounces 3.00 cm above and below the equilibrium position, the total displacement is 2 * 3.00 cm = 6.00 cm = 0.06 m.
The maximum potential energy can be calculated using the equation:
PE_max = 0.5 * k * x^2,
where k is the force constant of the spring and x is the maximum displacement.
Substituting the given values:
PE_max = 0.5 * 0.9 N/m * (0.06 m)^2
= 0.00108 J
According to the conservation of mechanical energy, this potential energy is converted into kinetic energy when the object reaches its maximum velocity.
Therefore, the kinetic energy at maximum velocity is equal to the potential energy:
KE_max = 0.00108 J
In scientific notation, KE_max ≈ 1.08 x 10^(-3) J.
Therefore, the object has approximately 1.08 x 10^(-3) J of kinetic energy at its maximum velocity.
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The A string on a violin has a fundamental frequency of a40 Hz. The length of the vibrating portion is 30.4 cm and has a mass of 0.342 g. Under what tension must the string be placed?
Answer: The tension in the A string of the violin must be placed under 263.7 N of tension.
The A string on a violin has a fundamental frequency of a 440 Hz.
To find the tension (T) in a string: T = (m * v²) / L
Where: m = the mass of the string, L = the length of the vibrating portion, v = the speed of the wave. The speed of the wave is given by the formula: v = √(T/μ)
Where T is the tension in the string and μ is the linear density of the string. To calculate the linear density of the string, we use the formula: μ = m/L
Fundamental frequency, f = 440 Hz
Length of the vibrating portion, L = 30.4 cm = 0.304 m
Mass of the string, m = 0.342 g = 0.000342 kg.
Using the frequency and the length of the vibrating portion, we can find the speed of the wave:
v = f * λλ
= 2L = 2(0.304 m)
= 0.608 mv
= (440 Hz)(0.608 m)
= 267.52 m/s.
Now, we can find the tension in the string:
T = (m * v²) / L
T = (0.000342 kg * (267.52 m/s)²) / 0.304 m
T ≈ 263.7 N.
Therefore, the tension in the A string of the violin must be placed under 263.7 N of tension.
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A high-voltage line operates at 500 000 V-rms and carries an rms current of 500 A. If the resistance of the cable is 0.50Ω/km, what is the resistive power loss over 200 km of the high-voltage line?
A.
500 kW
B.
25 Megawatts
C.
250 Megawatts
D.
1 Megawatt
E.
2.5 Megawatts
The resistive power loss over 200 km of the high-voltage line is 250 Megawatts. It corresponds to option C.
To calculate the resistive power loss, we need to determine the total resistance of the cable and then use the formula [tex]\text{P}=\text{I}^{2}\text{R}[/tex], where P is the power loss, I is the rms current, and R is the total resistance.
Given that the resistance of the cable is 0.50Ω/km, the total resistance for 200 km can be calculated as follows:
Total Resistance = (Resistance per kilometer) × (Total distance)
[tex]\text{R}=0.50\times200\\\text{R}=100\Omega[/tex]
Resistive power refers to the power loss or dissipation that occurs in a circuit or system due to the resistance of its components. It is the power that is converted into heat as electric current flows through a resistive element. Now, we can calculate the resistive power loss: Power Loss = (rms current)^2 × Total Resistance
[tex]\text{Power Loss}=\text{rms current}^2\times \text{total resistance}\\\\text{P}=500^{2}\times100\\\text{P}=250000\ \text{W}\\\text{P}=250\ \text{Megawatt}[/tex]
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Analyse the stick diagram as shown in Figure Q2(b). (i) Transform the stick diagram into the equivalent schematic circuit at transistor level. (10 marks) (ii) Determine the Boolean equation representing the output Y. (4 marks) Figure Q2(b)
The above schematic circuit diagram is the equivalent schematic circuit at transistor level.
The Boolean equation representing the output Y is X + Z.
(i) Transformation of stick diagram into an equivalent schematic circuit at transistor level
The stick diagram given above represents the schematic diagram of the given Boolean expression using only MOS transistors as per the design rules. The stick diagram can be transformed into the equivalent schematic circuit at transistor level as shown below:
The above schematic circuit diagram is the equivalent schematic circuit at transistor level.
(ii) Determination of Boolean equation representing the output Y Boolean equation can be formed by observing the schematic circuit diagram obtained from the stick diagram.
The output of the given circuit diagram is represented by the output terminal Y which is labelled in the circuit diagram obtained above. The output Y is formed by OR operation of the two input terminals X and Z as seen in the diagram. Therefore the Boolean equation representing the output Y is given as:
Y = X + Z.
The Boolean equation representing the output Y is X + Z.
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What is the energy of a photon that has the same wavelength as a 100-eV electron?
1) 100 eV
2) 10,000 eV
3) 1000 eV
4) 200 eV
5) 50 eV
The energy of the photon with the same wavelength as a 100-eV electron is:E = (hc)/(λ) = (1240 eV nm)/(12.4 pm) = 100 eVThus, the energy of the photon is 100 eV
The correct answer is option 1) 100 eV.Explanation:A photon is a massless particle that is a quantum of light. Its energy and wavelength are related through the equation:λ = hc/Ewhereλ = wavelength of the photonh = Planck's constantc = speed of lightE = energy of the photonAn electron with an energy of 100 eV will have a wavelength ofλ = h/(mv)where m is the mass of the electron and v is its velocity.
Using the De Broglie equation, we know that the wavelength of the electron isλ = h/(mv)Given that the energy of the photon is equal to the energy of the electron, we can equate the two expressions above:λ = hc/EEquating both equations, we get:hc/E = h/(mv)E = (hc)/(λ)Therefore, the energy of the photon with the same wavelength as a 100-eV electron is:E = (hc)/(λ) = (1240 eV nm)/(12.4 pm) = 100 eVThus, the energy of the photon is 100 eV.
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An oscillating LC circuit consisting of a 1.3 nF capacitor and a 4.0 mH coil has a maximum voltage of 3.8 V. What are (a) the maximum charge on the capacitor, (b) the maximum current through the circuit, (c) the maximum energy stored in the magnetic field of the coil? (a) Number 4.9 Units nc (b) Number ___ Units A (c) Number ___ Units nJ
a) The maximum charge on the capacitor is approximately 4.94 nC.
b) The maximum current through the circuit is approximately 0.043 A.
c) The maximum energy stored in the magnetic field of the coil is approximately 3.49 μJ.
(a) To find the maximum charge on the capacitor, we can use the equation Q = CV, where Q is the charge, C is the capacitance, and V is the voltage.
C = 1.3 nF = 1.3 × 10^(-9) F
V = 3.8 V
Substituting these values into the equation, we have:
Q = (1.3 × 10^(-9) F) × (3.8 V) = 4.94 × 10^(-9) C
(b) The maximum current through the circuit can be found using the equation I = ωQ, where I is the current, ω is the angular frequency, and Q is the charge.
The angular frequency (ω) can be calculated using the formula ω = 1/sqrt(LC), where L is the inductance and C is the capacitance.
L = 4.0 mH = 4.0 × 10^(-3) H
C = 1.3 nF = 1.3 × 10^(-9) F
Substituting these values into the formula, we have:
ω = 1/sqrt((4.0 × 10^(-3) H) × (1.3 × 10^(-9) F)) ≈ 8.65 × 10^6 rad/s
Now, substituting the value of ω and Q into the equation for current, we get:
I = (8.65 × 10^6 rad/s) × (4.94 × 10^(-9) C) ≈ 4.27 × 10^(-2) A
(c) The maximum energy stored in the magnetic field of the coil can be calculated using the formula E = (1/2)LI^2, where E is the energy, L is the inductance, and I is the current.
L = 4.0 mH = 4.0 × 10^(-3) H
I = 0.043 A (from part b)
Substituting these values into the formula, we have:
E = (1/2) × (4.0 × 10^(-3) H) × (0.043 A)^2 ≈ 3.49 × 10^(-6) J
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An object of mass m is suspended from a spring whose elastic constant is k in a medium that opposes the motion with a force opposite and proportional to the velocity. Experimentally the frequency of the damped oscillation has been determined and found to be √3/2 times greater than if there were no damping.
Determine:
a) The equation of motion of the oscillation.
b) The natural frequency of oscillation
c) The damping constant as a function of k and m
The equation of motion of the oscillation is m * d^2x/dt^2 + (c/m) * dx/dt + k * x = 0.The natural frequency of oscillation is 4km - 3k - c^2 = 0.The damping constant is = ± √(4km - 3k)
a) To determine the equation of motion for the damped oscillation, we start with the general form of a damped harmonic oscillator:
m * d^2x/dt^2 + c * dx/dt + k * x = 0
where:
m is the mass of the object,
c is the damping constant,
k is the elastic constant of the spring,
x is the displacement of the object from its equilibrium position,
t is time.
To account for the fact that the medium opposes the motion with a force opposite and proportional to the velocity, we include the damping term with a force proportional to the velocity, which is -c * dx/dt. The negative sign indicates that the damping force opposes the motion.
Therefore, the equation of motion becomes:
m * d^2x/dt^2 + c * dx/dt + k * x = -c * dx/dt
Simplifying this equation gives:
m * d^2x/dt^2 + (c/m) * dx/dt + k * x = 0
b) The natural frequency of oscillation, ω₀, can be determined by comparing the given frequency of damped oscillation, f_damped, with the frequency of undamped oscillation, f_undamped.
The frequency of damped oscillation, f_damped, can be expressed as:
f_damped = (1 / (2π)) * √(k / m - (c / (2m))^2)
The frequency of undamped oscillation, f_undamped, can be expressed as:
f_undamped = (1 / (2π)) * √(k / m)
We are given that the frequency of damped oscillation, f_damped, is (√3/2) times greater than the frequency of undamped oscillation, f_undamped:
f_damped = (√3/2) * f_undamped
Substituting the expressions for f_damped and f_undamped:
(1 / (2π)) * √(k / m - (c / (2m))^2) = (√3/2) * (1 / (2π)) * √(k / m)
Squaring both sides and simplifying:
k / m - (c / (2m))^2 = (3/4) * k / m
k / m - (c / (2m))^2 - (3/4) * k / m = 0
Multiply through by 4m to clear the fractions:
4km - c^2 - 3k = 0
Rearranging the equation:
4km - 3k - c^2 = 0
We can solve this quadratic equation to find the relationship between c, k, and m.
c) The damping constant, c, as a function of k and m can be determined by solving the quadratic equation obtained in part (b). Rearranging the equation:
c^2 - 4km + 3k = 0
Using the quadratic formula:
c = ± √(4km - 3k)
Note that there are two possible solutions for c due to the ± sign.
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A 2400 cm³ container holds 0.15 mol of helium gas at 320°C. Part A How much work must be done to compress the gas to 1200 cm³ at constant pressure? Express your answer to two significant figures and include the appropriate units. W = __________ Value ___________ Units Part B How much work must be done to compress the gas to 1200³ cm at constant temperature? Express your answer to two significant figures and include the appropriate units. W = __________ Value ___________ Units
The work done to compress the gas to 1200 cm³ at constant pressure is 28.3 J, and the work done at constant temperature is -31.9 J.
Container volume, V1 = 2400 cm³
Amount of gas, n = 0.15 mol
Temperature, T = 320°C
Final volume, V2 = 1200 cm³
Part A: We can calculate the work done using the formula,
W = -P∆V
where,
∆V = V2 - V1
P is constant and can be calculated using the ideal gas law equation PV = nRT.
So, P = (nRT) / V1
Substitute the given values to calculate P.
P = (0.15 mol * 8.31 J/mol*K * 593 K) / 2400 cm³ = 0.0236 atm
Now, calculate the work done.
W = - (0.0236 atm) * (1200 cm³ - 2400 cm³) = 28.3 J
Part B: When the temperature is constant, use the following formula to calculate work done.
W = nRT ln(V2/V1)
where,
R is the universal gas constant
R = 8.31 J/mol*K
Substitute the given values to calculate work done.
W = (0.15 mol * 8.31 J/mol*K * 593 K) ln(1200 cm³ / 2400 cm³)
W = -31.9 J (to two significant figures)
Therefore, the work done to compress the gas to 1200 cm³ at constant pressure is 28.3 J, and the work done at constant temperature is -31.9 J.
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loop coincides with the wire. Calculate the magnitude of the force exerted on the loop
A loop coincides with the wire.
To calculate the magnitude of the force exerted on the loop, we can use the formula:
F = BILsinθ, where F is the magnitude of the force exerted on the loop, B is the magnetic field strength, I is the current flowing through the wire, L is the length of the loop, and θ is the angle between the magnetic field and the plane of the loop.
Since the loop coincides with the wire, the angle θ between the magnetic field and the plane of the loop is 0 degrees. Therefore, sinθ = sin0 = 0. So the formula simplifies to:
F = BIL x 0 = 0
The force exerted on the loop is zero.
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Get the equation for energy. Explain the physical meaning of
energy in cfd.
The equation for energy in the context of fluid dynamics, specifically in Computational Fluid Dynamics (CFD), is typically represented by the conservation of energy equation, also known as the energy equation or the first law of thermodynamics. The equation can be expressed as:
ρ * (du/dt + u * ∇u) = -∇p + ∇⋅(μ * (∇u + (∇u)^T)) + ρ * g + Q
where:
ρ is the density of the fluid
u is the velocity vector
t is time
∇u represents the gradient of velocity
p is the pressure
μ is the dynamic viscosity
g is the gravitational acceleration vector
Q represents any external heat source/sink
The physical meaning of energy in CFD is the total energy of the fluid system, which includes kinetic energy (associated with the motion of the fluid), potential energy (associated with the elevation of the fluid due to gravity), and internal energy (associated with the fluid's temperature and pressure). The energy equation describes how this total energy is conserved and transformed within the fluid system.
In CFD simulations, the energy equation plays a crucial role in modeling the energy transfer, heat transfer, and flow characteristics within the fluid. It helps in understanding how energy is distributed, dissipated, and exchanged within the fluid domain. By solving the energy equation numerically, CFD simulations can predict temperature profiles, flow patterns, heat transfer rates, and other important parameters that are essential for various engineering applications, such as designing efficient cooling systems, optimizing combustion processes, and analyzing thermal behavior in fluid flows.
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A heat engine operating between energy reservoirs at 20∘C∘C and 640 ∘C∘C has 30 %% of the maximum possible efficiency.
How much energy must this engine extract from the hot reservoir to do 1100 JJ of work?
Express your answer to two significant figures and include the appropriate units.
Answer: The engine must extract 67,000 J of energy from the hot reservoir to do 1100 J of work.
The expression for the efficiency of a heat engine operating between two energy reservoirs at temperatures T1 and T2 is;η = 1 - (T1/T2)
T1 = 20 ° C and T2 = 640 ° C.
Efficiency of 30% : η = 0.30 = 1 - (20/640)
Therefore, we can solve for the temperature T2 as follows: T2 = 20 / (1 - 0.30)(640) = 1228.57 K.
The efficiency :η = 1 - (20/1228.57) = 0.9836
Thus, we can use this efficiency to calculate the energy: QH that must be extracted from the hot reservoir to do 1100 J of work as follows:
W = QH(1 - η)1100 J
= QH(1 - 0.9836)
QH = 1100 / (1 - 0.9836)
= 67,000 J.
Therefore, the engine must extract 67,000 J of energy from the hot reservoir to do 1100 J of work
Answer: 67,000 J
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The New Horizons space probe flew by Pluto in 2015. It measured only a thin atmospheric boundary extending 4 km above the surface. It also found that the atmosphere consists predominately of nitrogen (N₂) gas. The work to elevate a single N₂ molecule to this distance is 5.7536 x 10⁻²³ J. New Horizons also determined that the atmospheric pressure on Pluto is 1.3 Pa at a distance of 3 km from the surface. What is the atmospheric density at this elevation? mN2 = 2.32 x 10⁻²⁶kg a. 6.99 x 10⁻⁴ kg/m³ b. 442 x 10⁻² kg/m³ c. 442 x 10⁻⁵ kg/m³ d. 6.99 x 10⁻¹ kg/m³
Answer: The correct option is a. 6.99 x 10⁻⁴ kg/m³.
Work to elevate a single N₂ molecule to this distance = 5.7536 x 10⁻²³ Jm
N2 = 2.32 x 10⁻²⁶kg
Pluto Atmospheric Pressure = 1.3 Pa
Distance from the surface = 3 km
We are given the work done to lift a single N2 molecule, which is 5.7536 x 10⁻²³ J.
Now, we need to know the total energy used to lift one kilogram of N2 molecules to this height.
Since the mass of one N2 molecule is 2.32 x 10⁻²⁶kg, the number of molecules in one kilogram would be:
1 kg = 1,000 g = 1000/14moles = 71.43 moles.
In one mole, there are 6.022 x 10²³ molecules.
Therefore, in 71.43 moles, the number of N₂ molecules would be:71.43 moles x 6.022 x 10²³ molecules per mole
= 4.29 x 10²⁶ molecules of N₂.
Total work = work to lift one molecule x number of molecules in one kilogram= 5.7536 x 10⁻²³ J/molecule x 4.29 x 10²⁶ molecules/kg= 2.466 x 10³ J/kg.
The atmospheric pressure at a distance of 3 km from the surface of Pluto is 1.3 Pa.
Using the ideal gas law, PV = nRT,
where P is pressure, V is volume, n is the number of moles, R is the gas constant and T is temperature.
The mass of one N₂ molecule, m. N₂ is given as 2.32 x 10⁻²⁶ kg.
Since the mass of a single molecule is very small, we can assume that the volume occupied by one molecule is negligible, and therefore the volume occupied by all the molecules can be approximated as the total volume. The number of moles of N₂ gas in 1 kg would be:1 kg = 1000 g / (28 g/mol) = 35.71 moles.
Therefore, the number of molecules would be: 35.71 moles x 6.022 x 10²³ molecules/mole
= 2.15 x 10²⁶ molecules of N₂. The volume occupied by all the N2 molecules in 1 kg would be:,
V = nRT/P
= (35.71 x 8.314 x 55)/(1.3)
= 1.53 x 10³ m³.
The density of N₂ gas in 1 kg would be:
p = m/V = 1/1.53 x 10³
= 6.54 x 10⁻⁴ kg/m³.
Therefore, the atmospheric density at this elevation is 6.54 x 10⁻⁴ kg/m³.
Answer: The correct option is a. 6.99 x 10⁻⁴ kg/m³.
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A wire in the shape of an " \( \mathrm{M} \) " lies in the plane of the paper as shown in the figure. It carries a current of \( 2.00 \mathrm{~A} \), flowing from points \( A \) to \( B \), to \( C \)
The magnetic field at point P, which is inside the wire's loop, will be directed into the page or downward.
The wire in the shape of an "M" lies in the plane of the paper as shown in the figure. It carries a current of 2.00 A, flowing from points A to B, to C.What will be the direction of the magnetic field at point P due to the current-carrying conductor in the figure?We can apply the right-hand thumb rule to find the direction of the magnetic field at point P due to the current-carrying conductor in the figure.
The right-hand thumb rule states that if the thumb of the right hand is pointed in the direction of the current, the fingers will wrap around the conductor in the direction of the magnetic field.So, the magnetic field lines will flow around the wire in a counter clockwise direction (from points B to C to A). As a result, the magnetic field at point P, which is inside the wire's loop, will be directed into the page or downward. Therefore, the answer is "into the page or downward."
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A free electron has a kinetic energy 14.7eV and is incident on a potential energy barrier of U =32.3eV and width w=0.032nm. What is the probability for the electron to penetrate this barrier (in %)?
The probability for a free electron with a kinetic energy of 14.7 eV to penetrate a potential energy barrier of 32.3 eV and width 0.032 nm is very low, approximately 0.003%.
In quantum mechanics, the transmission probability of a particle through a potential energy barrier is described by the phenomenon of quantum tunneling. The probability of tunneling depends on various factors, including the width and height of the barrier, as well as the energy of the particle.
To calculate the transmission probability, we can use the transmission coefficient formula. The transmission coefficient (T) is given by T = [tex](1 + (U/E))^-2w^{2}[/tex], where U is the height of the potential energy barrier, E is the kinetic energy of the electron, and w is the width of the barrier. Plugging in the values, we have T = [tex](1 + (32.3 eV / 14.7 eV))^{2}[/tex] * 0.032 nm.
Calculating this expression, we find T ≈ 0.00003, or 0.003% when expressed as a percentage. This means that there is a very low probability for the electron to penetrate the barrier, indicating that most of the electrons will be reflected back rather than passing through.
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Which has the greater density—1 kg of sand or 10 kg of sand?.
Explain
The density of 1 kg of sand and 10 kg of sand is the same because the ratio of mass to volume remains constant.
Density is defined as mass per unit volume. In this case, we are comparing the densities of 1 kg of sand and 10 kg of sand.
Assuming the sand is uniform, the density remains constant regardless of the amount of sand. This means that both 1 kg of sand and 10 kg of sand have the same density.
To understand why the density remains the same, let's consider the definition of density:
Density = Mass / Volume
In this scenario, we are comparing the densities of two different amounts of sand: 1 kg and 10 kg. The mass increases by a factor of 10, but the volume also increases by the same factor. Assuming the sand particles remain the same and there is no compaction or voids, the volume scales linearly with mass.
Therefore, the density of 1 kg of sand and 10 kg of sand is the same because the ratio of mass to volume remains constant.
In conclusion, both 1 kg of sand and 10 kg of sand have the same density since the increase in mass is accompanied by an equal increase in volume.
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A body of mass 5kg is connected by a light inelastic string which is passed over a fixed frictionless pulley to a moveable frictionless pulley of mass 1kg over which is wrapped another light inelastic string which connects masses 3kg and 2kg. Find 1) the acceleration of the masses.
2) the tensions in the strings in terms of g, the acceleration dey to gravity
(a) The acceleration of the masses is determined as 1.1 m/s² in the direction of the 5 kg mass.
(b) The tension in the string in terms of gravity is T = g.
What is the acceleration of the masses?(a) The acceleration of the masses is calculated by applying Newton's second law of motion.
F(net) = ma
where;
m is the massesa is the acceleration of the masses(5 kg x 9.8 m/s² ) - (1 kg + 3 kg )9.8 m/s² = ma
9.8 N = (5kg + 1 kg + 3 kg )a
9.8 = 9a
a = 9.8 / 9
a = 1.1 m/s² in the direction of the 5 kg mass.
(b) The tension in the string in terms of gravity is calculated as follows;
T = ( 5kg)g - (1 kg + 3 kg ) g
T = 5g - 4g
T = g
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Elon Bezos launches two satellites of different masses to orbit the Earth circularly on the same radius. The lighter satellite moves twice as fast as the heavier one. Your answer NASA astronauts, Kjell Lindgren, Pilot Bob Hines, Jessica Watkins, and Samantha Cristoforetti, are currently in the International Space Station, and experience apparent weightlessness because they and the station are always in free fall towards the center of the Earth. Your answer True or False Patrick pushes a heavy refrigerator down the Barrens at a constant velocity. Of the four forces (friction, gravity, normal force, and pushing force) acting on the bicycle, the greatest amount of work is exerted by his pushing force. Your answer One of the 79 moons of Jupiter is named Callisto. The pull of Callisto on * 2 points Jupiter is greater than that of Jupiter on Callisto.
1. True - Astronauts in the International Space Station experience apparent weightlessness because they and the station are always in free fall towards the center of the Earth.
2. False - The pushing force exerted by Patrick does not do the greatest amount of work when he pushes a heavy refrigerator at a constant velocity.
3. False - The pull of Callisto on Jupiter is not greater than that of Jupiter on Callisto.
1. True - Astronauts in the International Space Station (ISS) experience apparent weightlessness because they and the station are in a state of continuous free fall around the Earth. They are constantly accelerating towards the Earth's center due to gravity, creating the sensation of weightlessness.
2. False - When Patrick pushes a heavy refrigerator at a constant velocity, the work done by the pushing force is zero because the displacement of the refrigerator is perpendicular to the force. The force of gravity, friction, and the normal force exerted by the ground contribute to the work done in balancing the forces and maintaining a constant velocity.
3. False - According to Newton's third law of motion, the gravitational force between two objects is equal and opposite. The pull of Callisto on Jupiter is equal in magnitude to the pull of Jupiter on Callisto, as governed by the law of universal gravitation.
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An air parcel begins to ascent from an altitude of 1200ft and a
temperature of 81.8°F. It reaches saturation at 1652 ft. What is
the temperature at this height? The air parcel continues to rise to
22
Given information:An air parcel begins to ascent from an
altitude
of 1200ft and a temperature of 81.8°F.It reaches
saturation
at 1652 ft.Now we have to find the temperature at this height?
The air parcel continues to rise to 22To find the temperature of the air parcel at an altitude of 1652 ft, we need to use the adiabatic lapse rate.
Adabatic lapse
rate refers to the rate of decrease of temperature with altitude in the troposphere, which is approximately 6.5 °C (11.7 °F) per kilometer (or 3.57 °F per 1,000 feet) of altitude.
Let T1 = 81.8°F be the temperature at an altitude of 1200ftand T2 = temperature at an altitude of 1652 ftLet the lapse rate be -6.5°C/km (or -3.57 °F / 1000ft).
At a height difference of 452 ft (1652 - 1200), the temperature decreases by 2.94°F (0.53°C),T2 = T1 - (lapse rate x height difference)T2 = 81.8 - (3.57 x 0.452)T2 = 80.6°F.
Therefore, at an altitude of 1652 ft, the temperature of the air parcel is approximately 80.6°F.
Given an air parcel starting at an altitude of 1200 ft with a temperature of 81.8°F, it reaches saturation at an altitude of 1652 ft. It is required to find out the temperature of the air parcel at 1652 ft. It is also given that the
air parcel
continues to rise to an unknown height.The answer to this problem requires the use of the adiabatic lapse rate formula.
Adiabatic lapse rate is defined as the rate at which temperature decreases with an increase in altitude in the troposphere. The
standard adiabatic lapse rate
is 6.5°C per kilometer, or 3.57°F per 1000 feet of altitude.
Let T1 = 81.8°F be the temperature at an altitude of 1200 ft.
Let T2 be the temperature at an altitude of 1652 ft.Let the lapse rate be -6.5°C/km (or -3.57 °F / 1000ft).
The temperature at an altitude of 1652 ft can be calculated asT2 = T1 - (lapse rate x height difference)T2 = 81.8 - (3.57 x 0.452)T2 = 80.6°F.
Therefore, at an altitude of 1652 ft, the temperature of the air parcel is approximately 80.6°F.
The
temperature
of the air parcel at an altitude of 1652 ft is 80.6°F. The adiabatic lapse rate formula was used to determine the temperature at this height.
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The temperature at which an air parcel reaches saturation is known as the dew point temperature. To determine the temperature at 1652 ft, we need to use the temperature equation, which relates the temperature and altitude of an ascending air parcel.
First, let's determine the temperature lapse rate, which is the rate at which the temperature changes with altitude. This can vary depending on atmospheric conditions, but a typical value is around 3.6°F per 1000 ft.
Using this lapse rate, we can calculate the change in temperature from 1200 ft to 1652 ft.
Change in altitude = 1652 ft - 1200 ft = 452 ft
Change in temperature = lapse rate * (change in altitude / 1000)
Change in temperature = 3.6°F/1000 ft * 452 ft = 1.6272°F
Next, we subtract the change in temperature from the initial temperature of 81.8°F to find the temperature at 1652 ft.
Temperature at 1652 ft = 81.8°F - 1.6272°F = 80.1728°F
Therefore, the temperature at 1652 ft is approximately 80.17°F.
The temperature at 1652 ft is approximately 80.17°F.
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A grinding wheel is a uniform cylinder with a radius of 7.20 cm and a mass of 0.350 kg <3 of 3 Constants Calculate its moment of inertia about its center. Express your answer to three significant figures and include the appropriate units. Calculate the applied torque needed to accelerate it from rest to 1750 rpm in 6.80 s. Take into account a frictional torque that has been measured to slow down the wheel from 1500 rpm to rest in 62.0 s Express your answer to three significant figures and include the appropriate units.
Plugging in the given values, we have I = (1/2)(0.350 kg)(0.072 m)² = 0.055 kg·m². This is the moment of inertia of the grinding wheel about its center.
To calculate the applied torque (τ) needed to accelerate the wheel, we use the equation τ = Iα, where α is the angular acceleration. The initial angular velocity is 0 (since the wheel starts from rest), and the final angular velocity is (1750 rpm)(2π rad/min) = (1750)(2π/60) rad/s. The time taken (t) is 6.80 s. Using the formula α = (ω - ω₀)/t, where ω is the final angular velocity and ω₀ is the initial angular velocity, we can calculate the angular acceleration. Substituting the values into τ = Iα, we can find the applied torque.
The frictional torque (τ_friction) that slows down the wheel is also given by τ_friction = Iα, where α is the angular acceleration. The initial angular velocity is (1500 rpm)(2π/60) rad/s, the final angular velocity is 0 (since the wheel comes to rest), and the time taken is 62.0 s. Using the formula α = (ω - ω₀)/t, we can calculate the angular acceleration. Substituting the values into τ_friction = Iα, we can find the frictional torque.
The applied torque is the difference between the torque needed for acceleration and the frictional torque: τ_applied = τ - τ_friction.
By performing the calculations, taking into account the given values and equations, we can determine the applied torque needed to accelerate the wheel and the effect of the frictional torque.
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The gravitational acceleration at the mean surface of the earth is about 9.8067 m/s². The gravitational acceleration at points A and B is about 9.8013 m/s² and 9.7996 m/s², respectively. Determine the elevation of these points assuming that the radius of the Earth is 6378 km. Round-off final values to 3 decimal places.
The elevation of point A is 15.945 km and the elevation of point B is 14.715 km
The formula used in solving the problem is given below:
h = R[2ga/G - 1]
Where
h = elevation
R = radius of Earth
ga = gravitational acceleration at A or B in m/s2
G = gravitational constant
The values of ga are
ga = 9.8013 m/s² at point A
ga = 9.7996 m/s² at point B.
Substituting these values into the formula gives the elevation
hA = R[2(9.8013)/9.8067 - 1]
= R[1.0025 - 1]
= R(0.0025)
hB = R[2(9.7996)/9.8067 - 1]
= R[1.0023 - 1]
= R(0.0023)
Thus the elevation of point A is 6378 km x 0.0025 = 15.945 km.
The elevation of point B is 6378 km x 0.0023 = 14.715 km (rounded to 3 decimal places).
Therefore, the elevation of point A is 15.945 km and the elevation of point B is 14.715 km (rounded to 3 decimal places).
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(a) Two point charges totaling 8.00μC exert a repulsive force of 0.300 N on one another when separated by 0.567 m. What is the charge ( in μC ) on each? smallest charge xμC μC (b) What is the charge (in μC ) on each if the force is attractive? smallest charge « μC largest charge μC
a)The charge on each particle in both cases is 4.00 μC and b) -1.86 x 10⁻⁶ C, respectively.
(a) Two point charges totaling 8.00μC exert a repulsive force of 0.300 N on one another when separated by 0.567 m. What is the charge (in μC) on each?The force between two point charges q1 and q2 that are separated by distance r is given by:F = (1/4πε) x (q1q2/r²)Here, ε = 8.85 x 10⁻¹² C²/Nm², q1 + q2 = 8.00 μC, F = 0.300 N, and r = 0.567 m.Therefore,F = (1/4πε) x [(q1 + q2)²/r²]0.300 = (1/4πε) x [(8.00 x 10⁻⁶)²/(0.567)²]q1 + q2 = 8.00 μCq1 = (q1 + q2)/2, q2 = (q1 + q2)/2Therefore,q1 = q2 = 4.00 μC.
(b) What is the charge (in μC) on each if the force is attractive?When the force is attractive, the charges are opposite in sign. Let q1 be positive and q2 be negative. The force of attraction is given by:F = (1/4πε) x (q1q2/r²)Therefore,F = (1/4πε) x [(q1 - q2)²/r²]0.300 = (1/4πε) x [(q1 - (-q1))²/(0.567)²]q1 = (0.300 x 4πε x (0.567)²)¹/² = 1.86 x 10⁻⁶ Cq2 = -q1 = -1.86 x 10⁻⁶ C. Thus, the charge on each particle in both cases is 4.00 μC and -1.86 x 10⁻⁶ C, respectively.
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