an electromagnetic wave has a frequency of 7.57 x 1014 hz, what is its wavelength? to what part of the spectrum does this wave belong?

The index of refraction must be 2, which is not physically possible, for a converging lens with two curved surfaces.

The central length of a combining focal point with two bended surfaces is given by the recipe 1/f = (n-1)[(1/R1) - (1/R2)], where f is the central length, n is the record of refraction, R1 and R2 are the radii of shape of the two surfaces. For this situation, the two surfaces have a span of shape of 10 cm, and the central length is likewise 10 cm. Accordingly, we can substitute these qualities into the equation and tackle for n:

1/10 = (n-1)[(1/10) - (1/10)]

n-1 = 1

n = 2

Hence, the record of refraction should be 2. It's quite significant that this worth isn't truly workable for a material to have, as the refractive file of any material should be more noteworthy than or equivalent to 1. Nonetheless, this outcome recommends that the focal point being referred to has an uncommon shape or material properties.

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The electron density between the 1s and 2s orbitals is indeed high at the nucleus, but the radial probability density drops to zero at the nucleus. This phenomenon is known as the "electron density paradox".

To understand why this is true, we need to consider the wave nature of electrons in an atom. According to the Schrödinger equation, the wave function of an electron is a probability amplitude that describes the probability of finding an electron in a particular location in space. The radial probability density is the probability of finding an electron at a particular distance from the nucleus. In the case of the 1s orbital, the electron has a high probability of being found at the nucleus because the wave function has a large amplitude there. However, the probability density drops to zero at the nucleus because the electron is a wave that must obey the laws of quantum mechanics, including the Heisenberg uncertainty principle. This principle states that it is impossible to determine both the position and momentum of an electron with absolute precision.

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Answer:

37.5 meters

Explanation:

We can use the equation:

v² - u² = 2ax

where v is the final velocity, u is the initial velocity, a is the acceleration, and x is the distance travelled during the acceleration.

We are given:

u = 5.5 m/s

v = 11.0 m/s

a = 1.2 m/s²

Substituting these values into the equation, we get:

11.0² - 5.5² = 2 × 1.2 × x

Simplifying the equation, we get:

120.25 - 30.25 = 2.4x

90 = 2.4x

x = 37.5 meters

Therefore, the distance traveled by the cyclist during the acceleration is 37.5 meters (to 2 significant figures).

The radius of the orbit is approximately 5.153 x [tex]10^{-2}[/tex] m or 5.15 cm.

To find the radius of the circular orbit of a proton with kinetic energy 1.0 x [tex]10^7[/tex] eV moving in a magnetic field of strength 5.0 x [tex]10^{-5}[/tex] T, follow these steps:
Convert the kinetic energy from electron volts (eV) to Joules (J) using the conversion factor

1 eV = 1.602 x [tex]10^{-19}[/tex] J.
Kinetic energy (J) = 1.0 x [tex]10^{7}[/tex] eV x 1.602 x [tex]10^{-19}[/tex] J/eV = 1.602 x [tex]10^{-12}[/tex] J
Calculate the velocity of the proton using the kinetic energy formula:
Kinetic energy = (1/2) x mass x [tex]velocity^2[/tex]
Solve for velocity:
Velocity = [tex]\sqrt{(2 x Kinetic energy / mass)}[/tex]
The mass of a proton is 1.67 x [tex]10^{-27}[/tex] kg

so:
Velocity = [tex]\sqrt{(2 \times 1.602 \times 10^{-12} J / 1.67 \times 10^{-27} kg)}[/tex]

Velocity  = 2.478 x [tex]10^{7}[/tex] m/s
Determine the charge of a proton, which is 1.602 x [tex]10^{-19}[/tex] C.
Calculate the radius of the orbit using the formula:
Radius = (mass x velocity) / (charge x magnetic field strength)
Radius = (1.67 x [tex]10^{-27}[/tex] kg x 2.478 x [tex]10^{7}[/tex] m/s) / (1.602 x[tex]10^{-19}[/tex]C x 5.0 x [tex]10^{-5}[/tex] T) = 5.153 x [tex]10^{-2}[/tex] m
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The water molecules move less quickly because some of the energy from the water was used to accelerate the metal. The water's temperature drops as a result of this.

The metal will eventually chill while the water warms up. The temperatures of the two items will eventually be equal. When this occurs, it is said that they are in thermal balance with one another.

Because heat moves from higher temperature to lower temperature when hot water and frigid water are combined, the mixture reaches an intermediate temperature. when hot water is followed by frigid water.

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Find the freezing point of the material and contrast it to that of other metals. Measure the metal's length and contrast it with the lengths of other metals. Also, contrast the metal's shape with those of other metals.

The air within the solar cooker receives energy from the water through the edges of the cook pot. The air outside solar cooker receives energy that is transferred from the gas within the solar cooker through sidewalls of the solar cooker.

Radiation is utilized to bake the s'mores in the hot cooker that is created in this exercise. When heat is transported without the presence of Sun and s'mores were in close proximity to one another  Electromagnetic waves that are passing through the air convey the heat.

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As per the given information, a certain non-conducing material has an index of refraction equal to 1.65. We have to determine the Brewster's angle.

The Brewster's angle is defined as the angle of incidence at which the reflected light becomes completely polarized perpendicular to the plane of incidence. It is also known as the polarization angle.

According to the Snell's law of refraction, μ = sin i/sin rWhere, μ is the refractive index of the material, i is the angle of incidence, and r is the angle of refraction.The Brewster's angle is given by the formula: tan β = μWhere, β is the Brewster's angleNow, we have the value of the refractive index, which is equal to 1.65.

Hence, we can use this value to calculate the Brewster's angle as follows: tan β = 1.65β = tan⁻¹(1.65)β = 58.45°Therefore, the Brewster's angle is 58.45° for the given non-conducting material with a refractive index of 1.65.

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For a classical wave, the following statements are true: b,c and the statements a and d are false.

a. Frequency is proportional to velocity: This statement is not true. Frequency and velocity are independent properties of a wave. The frequency of a wave refers to the number of oscillations or cycles per unit time, while the velocity of a wave refers to the speed at which the wave propagates.

b. Wavelength is proportional to frequency: This statement is partially true. In a classical wave, the wavelength is inversely proportional to the frequency. This means that as the frequency increases, the wavelength decreases, and vice versa.

c. Energy is proportional to the amplitude: This statement is true. The amplitude of a wave refers to the maximum displacement of the wave from its equilibrium position. The energy of a wave is proportional to the square of its amplitude. This means that as the amplitude increases, the energy carried by the wave also increases.

d. Energy is proportional to wavelength: This statement is not true. The energy of a classical wave is proportional to its frequency, not its wavelength. This is expressed by the formula E = hf, where E is the energy of the wave, h is Planck's constant, and f is the frequency of the wave.

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The efficiency of a subject on a treadmill can be calculated using the ratio of work output over work input. In this case, the work output is 117 W and the work input is 2.12 L/min of Oxygen.

The efficiency can be calculated as (117 W) / (2.12 L/min) * 100, which is approximately 55%. This means that 55% of the energy the subject is consuming is being used for work output and the remaining 45% is being used for other purposes such as heat loss or respiration. In other words, the subject is using 55% of their energy efficiently to produce work. In conclusion, the efficiency of the subject on a treadmill is 55%.

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A coil has an inductance of 7.00 mh, and the current in it changes from 0.200 a to 1.50 a in a time interval of 0.150 s. The magnitude of the average induced emf in the coil during this time interval is 0.72 V.

What is induced EMF, The induced EMF (electromotive force) is the voltage that is generated when a magnetic field passes through a conductor or is generated when a conductor moves through a magnetic field.

The formula for the magnitude of the average induced EMF in a coil is given by the following formula;

E = L (ΔI / Δt)

Where L is the inductance of the coil, ΔI is the change in current, and Δt is the time interval.

Substituting the given values,

L = 7.00 mH = 7.00 × 10-3 HΔI

= 1.50 A − 0.200 A

= 1.30 AΔt

= 0.150

sE = L (ΔI / Δt) = (7.00 × 10-3 H) (1.30 A / 0.150 s)E

= 0.072 V

therefore, The magnitude of the average induced EMF in the coil during this time interval is 0.72 V.

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In a single-slit experiment, when the slit width is decreased, the first two minima in the diffraction pattern will spread out and move further away from the central maximum.

In a single-slit experiment, when light passes through a narrow slit, it diffracts and creates a pattern of bright and dark fringes on a screen placed behind the slit. The location of the fringes depends on the wavelength of the light, the distance between the slit and the screen, and the width of the slit.

1. The single-slit experiment is based on the principle of diffraction, where light passing through a narrow slit creates an interference pattern.
2. The diffraction pattern consists of a central maximum surrounded by alternating bright and dark fringes, called maxima and minima, respectively.
3. The positions of the minima are determined by the relationship: a * sin(θ) = m * λ, where 'a' is the slit width, 'θ' is the angle between the central maximum and the minima, 'm' is the order of the minima (1, 2, 3, etc.), and 'λ' is the wavelength of the light.
4. As the slit width 'a' decreases, the value of sin(θ) must increase to maintain equality, which means the angle 'θ' increases as well.
5. With an increased angle 'θ', the first two minima in the diffraction pattern move further away from the central maximum, causing the pattern to spread out.

Therefore, as the slit width is decreased in a single-slit experiment, the distance between the first two minima in the diffraction pattern will increase.

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When the solar panel equipment arrives on site, the mounting hardware needs to be stored close to the work area because it will be installed first. Therefore the correct option is option A.

Mounting hardware, such as brackets and rails, is used to secure the solar panels to the roof or ground, and it needs to be installed before the panels can be mounted.

Until the mounting hardware is completed, the solar panels, inverters, and batteries can be kept somewhere else.

The solar panels themselves, inverters, and batteries can be stored elsewhere until the mounting hardware is installed. Therefore the correct option is option A.

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Standing waves can be produced in systems that have boundaries or endpoints that reflect or absorb waves. The nature of the boundary conditions determines the types of standing waves that can be produced.

a. Closed-closed: In a closed-closed system, both ends of the system are fixed and do not allow any displacement of the medium. This means that there is no flow of the medium at either end. An example of a closed-closed system is a string that is fixed at both ends.

b. Open-open: In an open-open system, both ends of the system are free to move, allowing the medium to flow in and out. This means that there is maximum displacement of the medium at both ends. An example of an open-open system is an air column that is open at both ends.

c. Closed-open: In a closed-open system, one end of the system is fixed and the other end is free to move, allowing the medium to flow in and out. This means that there is maximum displacement of the medium at the open end and no displacement at the closed end. An example of a closed-open system is a pipe that is closed at one end and open at the other.

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Answer: The buoyancy required for it to float stationary in the water is 49000

Explanation:

The charge stored in the capacitor is 36 coulombs.

When the voltage across a capacitor is 12 volts and its capacitance is 3 farads, the amount of charge stored in it is 36 coulombs.

What is voltage?

The voltage is the electric potential difference between two points in a circuit, or it may be the driving force that causes current to flow. The charge q stored in a capacitor with a capacitance C when a voltage V is applied is given by: q = CV, where q is the charge stored in the capacitor, C is the capacitance of the capacitor, and V is the voltage applied to the capacitor. A 3-farad capacitor that is charged to 12 volts stores 36 coulombs of charge. Therefore, 36 coulombs of charge is stored in it.

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The power delivered by both secondary coils A and B is the same (P_A = P_B), the comparison shows that the power delivered by secondary coil A is equal to the power delivered by secondary coil B.

Transformer A has a primary coil with 400 turns and a secondary coil with 200 turns. Transformer B has a primary coil with 400 turns and a secondary coil with 800 turns. The same current and voltage are delivered to the primary coil of both transformers, and the secondary coils are connected to identical circuits.

Step 1: Calculate the turns ratio for each transformer.
Transformer A:

Turns ratio (N) =

[tex]= N_{secondary} / N_{primary[/tex]

= 200 turns / 400 turns

= 0.5
Transformer B:

Turns ratio (N)

= [tex]N_{secondary} / N_{primary[/tex]

= 800 turns / 400 turns

= 2

Step 2: Determine the voltage and current ratios.
For transformers, the voltage ratio (V) and the current ratio (I) are inversely proportional to the turns ratio.
Transformer A:

Voltage ratio (V_A)

[tex]= N_A = 0.5,[/tex]

Current ratio (I_A)

[tex]= 1 / N_A = 2[/tex]
Transformer B:

Voltage ratio (V_B)

[tex]= N_B = 2,[/tex]

Current ratio (I_B)

[tex]= 1 / N_B = 0.5[/tex]

Step 3: Calculate the power delivered by the secondary coils.
Power (P) = Voltage (V) × Current (I)
For Transformer A:

[tex]P_A = V_A \times I_A \\= 0.5 \times 2 = 1[/tex](relative value)
For Transformer B:

[tex]P_B = V_B \times I_B \\= 2 \times 0.5 = 1[/tex](relative value)

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Sirius, the brightest star in the night sky, is actually a binary star system. Sirius A is a main-sequence star, and Sirius B is a white dwarf. Nearly all the visible light we see from Sirius comes from Sirius A. But when we photograph the system with X-ray light, as shown , Sirius B is the brighter of the two stars because As a white dwarf, Sirius B is much hotter than Sirius A and thus emits more X-rays.

Sirius is a binary star system made up of two stars called Sirius A and Sirius B.  Sirius A is the larger and more massive star, while Sirius B is a smaller, denser, and hotter white dwarf. Sirius A is a main-sequence star, similar to our Sun, while Sirius B is the remnant of a star that has exhausted all of its nuclear fuel and has collapsed down to a dense core.

Although Sirius A is much brighter than Sirius B in visible light, as mentioned, Sirius B is actually the brighter of the two stars when observed in X-ray light. This is because as a white dwarf, Sirius B has a much smaller surface area than Sirius A, but is much hotter and therefore emits more energy per unit area.

The high temperature of Sirius B's surface causes it to emit a significant amount of X-ray radiation, which is why it appears brighter than Sirius A in X-ray images of the system.

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The probable question may be:

sirius, the brightest star in the night sky, is actually a binary star system. sirius a is main-sequence star and sirius b is a white dwarf. nearly all the visible light we see from sirius comes from sirius a. but when we photograph the system with x-ray light, as shown here, sirius b is the brighter of the two stars. why?

Therefore, a rock dropped in Valles Marineris on Mars would fall approximately 150.255 meters after 9 seconds.

The majority of scientists concur that Valles Marineris is a sizable tectonic "crack" that formed in the Martian crust as the planet cooled, was influenced by the rising crust in the Tharsis area to the west, and then widened by erosional forces.

After 9 seconds, an object dumped on Mars would have travelled the distance indicated by the following formula:

d = 0.5 * g * t²

When we enter the specified numbers into the formula, we obtain:

d = 0.5 * 3.71 m/s² * (9 s)²

d = 0.5 * 3.71 m/s² * 81 s²

d = 150.255 m

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The total amount of energy radiated by the accretion disk during the history of the black hole is approximately  6.44 x 10^53 Joules. The average luminosity of the accretion disk  is approximately 2.04 x 10^37 Watts.

The average luminosity of the accretion disk is slightly lower than the luminosity of the Milky Way.

To determine the total amount of energy radiated by the accretion disk during the history of the black hole, we can use the mass-energy equivalence formula, E=mc^2, where E is the energy, m is the mass, and c is the speed of light (approx. 3x10^8 m/s).

First, we find 10% of the black hole's mass: 0.1 x 36 million solar masses = 3.6 million solar masses. Then, we convert this mass to kilograms: 3.6 million solar masses x (1.989 x 10^30 kg/solar mass) ≈ 7.16 x 10^36 kg.

Next, we calculate the energy: E = (7.16 x 10^36 kg) x (3 x 10^8 m/s)^2 ≈ 6.44 x 10^53 Joules.
To find the average luminosity of the accretion disk, we divide the total energy by the time it radiated that energy: 6.44 x 10^53 Joules / (10 billion years x 3.1536 x 10^7 s/year) ≈ 2.04 x 10^37 Watts.

Comparing the average luminosity of the accretion disk to the luminosity of the Milky Way, which is around 3 x 10^37 Watts, we see that the average luminosity of the accretion disk is slightly lower than the luminosity of the Milky Way.

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The clay was tossed at a speed of 15 m/s. The system's overall momentum is still preserved, but its total kinetic energy is not.

In an isolated system, when two objects contact, the total momentum before and after the collision is equal, according to the law of conservation of momentum. This is because the momentum that is lost by one object and acquired by another is equivalent.

a. we can use the conservation of momentum equation:

m1v1 + m2v2 = (m1 + m2)v

m1 and v1 are the mass and starting velocity of the clay, m2 and v2 are the mass and initial velocity of the wooden block (which is immobile), respectively.

Substitute in the given values,

(0.10 kg) v1 + (0.10 kg) (0 m/s) = (0.10 kg + 0.10 kg) (15 m/s)

Solving for v1,

v1 = 15 m/s

b. The wooden block experiences force in the opposite direction of the motion as the bouncy ball bounces off of it. The block finally slows down and stops as a result of this. The wooden block is affected by the bouncing ball in such a way that it receives some of the ball's kinetic energy and moves in the opposite direction of the ball. The wooden block slows down and comes to a stop, while the ball continues to move because it bounces back and retains some of its kinetic energy.

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Answer:

L = 912 kg*m^2/s

Explanation:

L = mvr

L = 80kg(6m/s)(1.9m)

L = 912

The angular momentum of the person is 912 kg⋅m²/s.

The angular momentum (L) of an object is given by the product of its moment of inertia (I) and its angular velocity (ω). The moment of inertia of a point mass rotating about an axis at a distance r from the center is given by I = mr², where m is the mass of the object.

In this case, the person is sitting on the edge of a rotating platform at a distance of 1.9 m from the center. Therefore, the moment of inertia of the person-platform system is I = mr² = 80 kg × (1.9 m)² = 288.8 kg⋅m².

The tangential speed (v) of the person is given as 6.0 m/s. The angular velocity (ω) can be calculated as ω = v/r, where r is the distance from the center. Therefore, ω = 6.0 m/s ÷ 1.9 m = 3.16 rad/s.

Now, we can calculate the angular momentum of the person as L = Iω = (288.8 kg⋅m²) × (3.16 rad/s) = 912 kg⋅m²/s.

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The magnitude of the magnetic flux through each turn of the coil at the instant when the current is 3.60 A is 3.72×10⁻⁶ Wb.

The quantity of magnetic field traveling through a specific region or surface is measured by magnetic flux. It is described as the result of the area that the magnetic field travels through and its magnitude, where the area is perpendicular to the field's direction. The Weber is the magnetic flux SI unit (Wb)

Given that,

EMF = 18.0 mV = 0.018 V

N = 484 turns

(dΦ/dt) = 10.0 A/s

I = 3.60 A

The equation that relates the electromotive force (EMF) induced in a coil to the rate of change of magnetic flux is:

EMF = -N(dΦ/dt)

Rearranging the equation to solve for Φ:

Φ = -EMF/(N(d/dt))

   = -(0.018 V)/(484(10.0 ))

  = -3.72×10⁻⁶ Wb

The magnitude of the magnetic flux through each turn of the coil at the instant when the current is 3.60 A is 3.72×10⁻⁶ Wb.

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The acceleration of the system still depends only on the difference in mass between the two masses, but the direction of the net force has changed, so the direction of acceleration has also changed.

When the masses are different, the system accelerates in the direction of the heavier mass. According to Newton's second law, the net force acting on each mass is equal to its mass times its acceleration:

F1 = m1a

F2 = m2a

where F1 and F2 are the tensions in the rope on either side of the pulley. Since the rope is inextensible and the pulley is ideal, the tension in the rope is the same on both sides, so F1 = F2 = T, where T is the tension in the rope.

Substituting T for F1 and F2 in the above equations and subtracting the second equation from the first, we get:

m1a - m2a = T - T

(m1 - m2)a = 0

Since the tension cancels out, the acceleration of the system depends only on the difference in mass between the two masses.

If the masses are switched, then the direction of the net force changes, and the system accelerates in the direction of the new heavier mass. The net force acting on each mass is still given by the above equations, but the masses are switched, so m1 is now the smaller mass, and m2 is the larger mass.

Substituting T for F1 and F2 in the above equations and subtracting the first equation from the second, we get:

m2a - m1a = T - T

(m2 - m1)a = 0

Since the tension cancels out, the acceleration of the system still depends only on the difference in mass between the two masses.

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the change in entropy is: A- ΔS = Q/T = (105,000 J) / (273 K) = 384.6 J/K b-  ΔS = -Q/T = (-166,750 J) / (273 K) = -611.2 J/K,c- ΔS = -Q/T = (-209,300 J) / (373 K) = -560.4 J/K, d)- ΔS = Q/T = (1,128,500 J) / (373 K) = 3023.5 J/K

a) To calculate the change in entropy when heating 0.5 kg of solid ice from -100°C to 0°C, we can use the equation:

ΔS = Q/T

where Q is the heat transferred to the system and T is the temperature at which the heat transfer occurs. Since the system is receiving heat, the sign of ΔS will be positive.

The heat required to raise the temperature of the ice from -100°C to 0°C can be calculated as:

Q = mcΔT

where m is the mass of the ice, c is the specific heat capacity of ice, and ΔT is the change in temperature.

Using the values given, we get:

Q = (0.5 kg) (2100 J/kg°C) (100°C) = 105,000 J

Therefore, the change in entropy is:

ΔS = Q/T = (105,000 J) / (273 K) = 384.6 J/K

b)  Q = (0.5 kg) (333,500 J/kg) = 166,750 J

Therefore, the change in entropy is:

ΔS = -Q/T = (-166,750 J) / (273 K) = -611.2 J/K

c) Q = (0.5 kg) (4186 J/kg°C) (100°C) = 209,300 J

Therefore, the change in entropy is:

ΔS = -Q/T = (-209,300 J) / (373 K) = -560.4 J/K

d) Q = (0.5 kg) (2,257,000 J/kg) = 1,128,500 J

Therefore, the change in entropy is:

ΔS = Q/T = (1,128,500 J) / (373 K) = 3023.5 J/K

e) The change in entropy for vaporizing water (3023.5 J/K) is larger

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With a temperature change of 31°C, the mercury column's height changes by 0.0106 cm.

As the temperature rises, the mercury will expand in the capillary tube, increasing its volume. V = VT = (1.8*10-4 (oC)-1)(0.100 cm3)(20 oC) = 3.6*10-4 cm3 = 0.36 mm3, or the formula V = VT.

[tex]A1 * h1 = A2 * h2[/tex]

[tex]A1 = πr1^2 = π(0.005 cm/2)^2 = 7.85 × 10^-5 cm^2[/tex]

The cross-sectional area of the bulb is:

[tex]A2 = πr2^2 = π(0.31 cm/2)^2 = 0.0755 cm^2[/tex]

[tex]ΔV = V0 * β * ΔT[/tex]

[tex]V0 = A1 * h1[/tex]

[tex]h2 = (A1/A2) * h1 + (ΔV/A2)[/tex]

[tex]h2 = (7.85 × 10^-5 cm^2)/(0.0755 cm^2) * h1 + (0.000182 (°C)^-1 * 31°C *[/tex] 7.85 × [tex]10^-5 cm^2)/(0.0755 cm^2)[/tex]

[tex]h2 = 0.0106 h1 + 0.00000122 cm[/tex]

[tex]Δh = h2 - h1 = 0.0106 h1 + 0.00000122 cm - h1 = 0.0106 cm[/tex]

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When 20 pillars of the same height are erected along the boundary of a circular stadium and the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, the total number of beams is 190.

There are 20 pillars, and each pillar can be connected to 16 non-adjacent pillars (i.e., every 2nd pillar). Hence, the total number of connections would be 20 × 16 = 320. However, each connection is counted twice since it connects two pillars. Therefore, we need to divide 320 by 2 to obtain the actual number of connections. Therefore, there are 160 beams that connect the top of the pillars.

However, we need to consider the beams that form the boundary of the stadium as well. Since there are 20 pillars, there are 20 beams that connect the adjacent pillars on the boundary. Therefore, the total number of beams would be 160 + 20 = 180. But we still need to consider the beams that connect the pillars at the center of the stadium, which would be the diameter of the circle on which the pillars are erected.

If the radius of the circle is r and the height of the pillars is h, then the length of the diameter would be 2r + h. Since there are 20 pillars, there would be 20 diameters. Hence, the total number of beams that connect the pillars at the center of the stadium would be 20 × (2r + h). Therefore, the total number of beams would be:180 + 20(2r + h)= 180+40r+20h. Thus, the total number of beams is 180 + 40r + 20h.

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The acceleration (α) of the center of the hoop: α = (mg * sin(θ)) / (2m) = (g * sin(θ)) / 2

The minimum coefficient of static friction (μmin) needed for the hoop to roll without slipping: μmin = α / (g * cos(θ)) = (g * sin(θ) / 2) / (g * cos(θ)) = tan(θ) / 2

The acceleration (α) of the center of the circular hoop can be determined by applying Newton's second law and the principle of conservation of angular momentum. Considering gravitational force (mg), normal force (N), and friction force (f) acting on the hoop, we can write:

mg * sin(θ) - f = m * α (1) (linear motion) and f * r = I * α/r (2) (angular motion)

For a hoop, the moment of inertia (I) is given by I = m * r^2. Substituting this into equation (2) and solving for f, we get:

f = m * α (3)

Now, substitute equation (3) into equation (1):

mg * sin(θ) - m * α = m * α

Rearranging the terms, we get the acceleration (α) of the center of the hoop:

α = (mg * sin(θ)) / (2m) = (g * sin(θ)) / 2

For the hoop to roll without slipping, the static friction (f) must be equal to the torque acting on the hoop. The minimum coefficient of static friction (μmin) can be determined using the frictional force equation:

f = μmin * N

Since the normal force (N) equals mg * cos(θ), the frictional force can be written as:

f = μmin * mg * cos(θ)

Now, substituting f = m * α from equation (3) into this equation, we get:

m * α = μmin * mg * cos(θ)

Rearranging the terms, we find the minimum coefficient of static friction (μmin) needed for the hoop to roll without slipping:

μmin = α / (g * cos(θ)) = (g * sin(θ) / 2) / (g * cos(θ)) = tan(θ) / 2

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The potential difference between the points at yi = -7cm and yf = 7cm in the given electric field is 1,050,000 V.

The electric field is given as E⃗ = (20,000i^ - 75,000j^) V/m. This means that the electric field is only in the y-direction, and its magnitude is constant throughout.

To find the potential difference between two points in the electric field, we can use the following formula:

ΔV = -∫E⃗ · dl⃗

where ΔV is the potential difference, E⃗ is the electric field, and dl⃗ is an infinitesimal displacement along the path between the two points. The negative sign arises because the electric field is acting in the opposite direction to the direction of motion along the path.

In this case, we can assume a path that moves from the point at yi = -7cm to the point at yf = 7cm, both of which lie on the y-axis. Since the electric field is only in the y-direction, we can assume that the path is also along the y-axis. Therefore, we can simplify the formula to:

ΔV = -∫E dy

where the integral is taken from yi = -7cm to yf = 7cm.

Substituting the electric field E⃗ and integrating, we get:

ΔV = -∫E dy = -∫(-75,000) dy (since the y-component of the electric field is -75,000 V/m)

ΔV = -[-75,000y]yi=-7cm to yf=7cm

ΔV = -[-75,000(7cm - (-7cm))] = 1,050,000 V

Therefore, the potential difference between the points at yi = -7cm and yf = 7cm in the given electric field is 1,050,000 V.

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Answer:

1) ≈ 0,11 A

2) 224 V

3) I added a photo of my drawing

4) ≈ 8,34 Ω

5) ≈ 0,94 A

6) ≈ 0,35 A

7) 0,94 A

Explanation:

1) Given:

R = 80 Ω

V = 9V

Find: I - ?

[tex]i = \frac{v}{r} = \frac{9}{80} ≈ 0.11[/tex]

.

2) Given:

I = 2 A

R1 = 100 Ω

R2 = 12 Ω

Find: V - ?

This circuit is connected in series

R = R1 + R2

R = 100 + 12 = 112 Ω

V = I × R

V = 2 × 112 = 224 V

.

3) I added a photo of my drawing

4) Given:

R1 = R2 = R3 = 82 Ω

R4 = 12 Ω

V = 9 V

Find: R (total) - ?

This circuit is connected in parallel

[tex]\frac{1}{r} = \frac{1}{r1} + \frac{1}{r2} + \frac{1}{r3} + \frac{1}{r4} [/tex]

Since R1 = R2 = R3, we can find the total resistance in these 3 resistors using this formula:

[tex]r = \frac{r1}{n} [/tex]

n - the number of resistors

r1 - the resistance of one resistor (when all resistors have the same resistance)

[tex]r(123) = \frac{82}{3} ≈ 27.33[/tex]

Now, let's find the remaining resistance:

[tex] \frac{1}{r(total)} = \frac{1}{27.33} + \frac{1}{12} = \frac{437}{3644} [/tex]

[tex]r(total) ≈8.34[/tex]

.

5) Given:

R1 = R2 = 50 Ω

R3 = 75 Ω

R4 = 45 Ω

V = 120V

Find: I - ?

R3 and R4 are connected in parallel

[tex] \frac{1}{r(34)} = \frac{1}{75} + \frac{1}{45} = \frac{8}{225} [/tex]

[tex]r(34) ≈28.13[/tex]

R1, R2 and R34 and conected in series

[tex]r(total) = 28.13 + 50 + 50 = 128.13[/tex]

[tex]i = \frac{v}{r} = \frac{120}{128.13} ≈0.94[/tex]

.

6) Given:

R3 = 75 Ω

Find: I3 - ?

First, we have to find the voltage across R3 and R4

In order to do that, first we have know what voltage is across R1 and R2 (since the resistance is these 2 resistors are the same, the voltage will also be the same):

V1 = V2 = I × R1 = 0,94 × 50 = 47 V

Then V4 = V5 (parallel conection) = 120 - 47 - 47 = 26 V

I3 = V3/R3 = 26/75 = 0,35 A

.

7) Given:

I (total) = 0,94 A

Find: I1, I2 - ?

I1 = I2 = 0,94 A, because the current in series connection stays the same in every part of the circuit