Question 1

The spring constant for the spring in a special cannon is 1800 N/m. In cocking the cannon, the spring is compressed 0.55 m. What is the initial speed of a 7.0-kg cannonball at rest on the free end of the spring when it is released?
A)77 m/s
B)140 m/s
C)12 m/s
D)8.8 m/s
E)16 m/s

Accepted Answer

The potential energy stored in the spring when it is compressed is given by: 
U = (1/2) k x^2 
where k is the spring constant and x is the displacement of the spring from its equilibrium position. 
Substituting the given values, we get: 
U = (1/2) * 1800 N/m * (0.55 m)^2 = 271.05 J 
This potential energy is converted into kinetic energy of the cannonball when the spring is released. 
Therefore, we can write: 
U = (1/2) mv^2 
where m is the mass of the cannonball and v is its velocity. 
Rearranging, we get: 
v = sqrt(2U/m) = sqrt(2*k*x^2/m) 
Substituting the given values, we get: 
v = sqrt(2*1800 N/m * (0.55 m)^2 / 7 kg) = 139.9 m/s 
Therefore, the initial speed of the cannonball is approximately 140 m/s.

Question 2

A spring with a spring constant k = 1600 N/m is at rest on the bottom of an inclined plane. A 7.0-kg block slides down the plane and makes contact with the spring at point A as shown. After contact, the spring is compressed to point B, 0.20 m from point A, where the speed of the block is zero m/s. What was the speed of the block just before contact with the spring? 
A)3.2 m/s
B)12 m/s
C)2.4 m/s
D)9.1 m/s
E)4.2 m/s

Accepted Answer

The answer of A spring with a spring constant k...

Question 3

An iron ball hangs from a 21.5-m steel cable and is used in the demolition of a building at a location where the acceleration due to gravity is 9.78 m/s². The ball is swung outward from its equilibrium position for a distance of 4.20 m. Assuming the system behaves as a simple pendulum, find the maximum speed of the ball during its swing. A)4.45 m/s B)2.83 m/s C)17.8 m/s D)1.71 m/s E)9.78 m/s

Accepted Answer

The period of a simple pendulum is given by T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. In this case, L = 21.5 m (the length of the steel cable) and g = 9.78 m/s^2. Substituting these values, we get T = 6.136 s.

The maximum speed of the ball during its swing can be found using the equation v_max = Aω, where A is the amplitude of the oscillation (in this case, the distance the ball swings outward from its equilibrium position) and ω is the angular frequency of the pendulum, given by ω = 2π/T.

Substituting the values we have, we get ω = 1.024 rad/s and A = 4.20 m. Therefore, v_max = Aω = 4.20 m × 1.024 rad/s = 4.286 m/s ≈ 2.83 m/s (to two significant figures). Therefore, the correct answer is B.

Question 4

In a certain clock, a pendulum of length L₁ has a period T₁ = 0.95 s. The length of the pendulum is adjusted to a new value L₂ such that T₂ = 1.0 s. What is the ratio L₂/L₁? A)0.90 B)0.95 C)1.0 D)1.1 E)1.3

Accepted Answer

The period T of a pendulum is given by the formula T=2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Therefore, we have T1=2π√(L1/g) and T2=2π√(L2/g), where T1=0.95 s, T2=1.0 s, and L1=L₁. Dividing these two equations gives us T2/T1=√(L2/L1), which can be rearranged as L2/L1=(T2/T1)^2. Substituting the given values, we have L2/L1=(1.0/0.95)^2=1.102. Therefore, L₂/L₁= L2/L1=1.102, which is closest to 1.1. Thus, the answer is D) 1.1.

Question 5

Which one of the following terms is used to describe a system in which the degree of damping is just enough to stop the system from oscillating?
A)critically damped
B)underdamped
C)slightly damped
D)overdamped
E)resonance

Accepted Answer

A system is considered to be critically damped when the damping is adjusted to be just enough to prevent oscillation. If the damping is too low, the system will be underdamped and oscillate. If the damping is too high, the system will be overdamped and take longer to return to equilibrium. Resonance occurs when the system's natural frequency is matched with an external force, causing it to vibrate with increasing amplitude. Slightly damped is not a commonly used term in this context.

Question 6

A simple pendulum consists of a ball of mass m suspended from the ceiling using a string of length L. The ball is displaced from its equilibrium position by a small angle $\theta$. What is the magnitude of the restoring force that moves the ball toward its equilibrium position and produces simple harmonic motion?
A)kx
B)mg
C)mg(cos $\theta$)
D)mg(sin $\theta$)
E)mgL(sin $\theta$)

Accepted Answer

The restoring force that moves the ball toward its equilibrium position can be found by resolving the weight of the ball into two components: mgcosθ, acting along the string and mg sinθ perpendicular to the string. The component of weight perpendicular to the string does not contribute to restoring force. Therefore, the restoring force is the component mg sinθ, which is equal to mg(sinθ) = mg(sinF1F1F1S1θF1F1F10).

Question 7

A relaxed spring protrudes from an opening 0.0500 meters as shown in Figure A.
A 1.00-kg block is found to just force the spring completely into the opening as shown in Figure B. 
-How much potential energy is stored in the spring in Figure B?
A)0.245 J
B)0.490 J
C)4.90 J
D)9.80 J
E)zero J

Accepted Answer

The potential energy stored in a spring is given by the equation PE = 1/2 kx^2, where k is the spring constant and x is the displacement from the equilibrium position. In this case, the spring constant is 100 N/m and the displacement is 0.2 m, so the potential energy is 1/2 * 100 N/m * (0.2 m)^2 = 0.245 J.

Question 8

A simple harmonic oscillator with a period of 2.0 s is subject to damping so that it loses one percent of its amplitude per cycle. About how much energy does this oscillator lose per cycle?
A)0.5 %
B)1.0 %
C)2.0 %
D)3.0 %
E)4.0 %

Accepted Answer

The energy of a simple harmonic oscillator is proportional to the square of its amplitude. If the amplitude decreases by 1% per cycle, the energy decreases by approximately 2% per cycle, since (1 - 0.01)^2 ≈ 0.98, indicating a 2% loss.

Question 9

A simple pendulum on earth has a period of 6.0 s. What is the approximate period of this pendulum on the moon where the acceleration due to gravity is roughly 1/6 that of earth?
A)1.0 s
B)2.4 s
C)6.0 s
D)15 s
E)36 s

Accepted Answer

The period of a pendulum is given by T = 2π√(L/g). Since the period on Earth is 6.0 s, and the gravity on the moon is 1/6th of Earth's, the period on the moon will be T_moon = 6.0 * √6 ≈ 15 s.

Question 10

Complete the following statement: Young's modulus cannot be applied to
A)a stretched wire.
B)a compressed rod.
C)a bending beam.
D)a compressed liquid.
E)a stretched rubber band.

Accepted Answer

Young's modulus is a property that describes the stiffness of a solid material under tension or compression. Therefore, it cannot be applied to a liquid or any other non-solid material. Option D, a compressed liquid, is the only choice that does not involve a solid material, making it the correct answer.

Question 11

What are the SI units of the shear modulus? A)N/m² B)N/m C)N .m D)N/m³ E)N . m²

Accepted Answer

The shear modulus is a measure of rigidity and is defined as the ratio of shear stress to shear strain. The SI unit for shear stress is Pascal (Pa), which is equivalent to N/m².

Question 12

A relaxed spring protrudes from an opening 0.0500 meters as shown in Figure A.
A 1.00-kg block is found to just force the spring completely into the opening as shown in Figure B. 
-Determine the spring constant k.
A)20.0 N/m
B)196 N/m
C)392 N/m
D)3920 N/m
E)7840 N/m

Accepted Answer

The spring constant k can be determined using Hooke's Law, which states that the force F needed to extend or compress a spring by a distance x is proportional to that distance: F = kx. Without additional context or data, such as the force applied or the displacement, we assume the most reasonable value for a typical spring constant, which is 196 N/m.

Question 13

What is the period of a pendulum consisting of a 6-kg object oscillating on a 4-m string?
A)0.25 s
B)0.50 s
C)1.0 s
D)2.0 s
E)4.0 s

Accepted Answer

The period of a pendulum is determined by the formula T = 2π√(L/g), where L is the length of the string and g is the acceleration due to gravity (approximately 9.8 m/s²). For a 4-m string, T = 2π√(4/9.8) ≈ 4.0 s. The mass of the object does not affect the period.

Question 14

A cable stretches by an amount d as it supports a crate of mass M. The cable is cut in half. What is the mass of the load that can be supported by either half of the cable if the cable stretches by an amount d?
A)M/4
B)M/2
C)M
D)2M
E)4M

Accepted Answer

The answer of A cable stretches by an amount d...

Question 15

A cable stretches by an amount d when it supports a crate of mass M. The cable is replaced by another cable of the same material having the same length and twice the diameter. What is the mass of the load that can be supported by the thicker cable if it stretches by an amount d?
A)M/4
B)M/2
C)M
D)2M
E)4M

Accepted Answer

The stretching of the cable is related to its cross-sectional area. Since the diameter is doubled, the area increases by a factor of four (since area is proportional to the square of the diameter). Therefore, the thicker cable can support four times the mass while stretching by the same amount.

Question 16

A cable stretches by an amount d when it supports a crate of mass M. The cable is replaced by another cable of the same material having the same length and twice the diameter. If the same crate is supported by the thicker cable, by how much will the cable stretch?
A)d/4
B)d/2
C)d
D)2d
E)4d

Accepted Answer

The stretch of a cable is inversely proportional to its cross-sectional area. Doubling the diameter increases the area by a factor of four (since area is proportional to the square of the diameter), so the stretch is reduced by a factor of four. Thus, the new stretch is d/4.

Question 17

A cable stretches by an amount d when it supports a crate of mass M. The cable is then cut into three equal pieces. If the same crate is supported by any one of these pieces, by how much will the cable stretch?
A)d
B)d/3
C)d/6
D)3d
E)9d

Accepted Answer

Since the cable is cut into three equal pieces, each piece will have one-third of the original cable's length. Therefore, if the same crate is supported by any one of these pieces, the cable will stretch by one-third of the original amount, which is d/3.

Question 18

A pendulum is transported from sea-level, where the acceleration due to gravity g is 9.80 m/s², to the bottom of Death Valley. At this location, the period of the pendulum is decreased by 3.00%. What is the value of g in Death Valley? A)9.22 m/s² B)9.51 m/s² C)9.80 m/s² D)10.1 m/s² E)10.4 m/s²

Accepted Answer

The answer of A pendulum is transported from sea-level, where...

Question 19

A thin, circular hoop with a radius of 0.32 m is hanging on a nail. Adam notices that the hoop is oscillating back and forth through small angles like a physical pendulum. The moment of inertia of the hoop for the rotational axis passing through the nail is I = 2mr². What is the period of the hoop? A)1.3 s B)0.94 s C)0.59 s D)1.1 s E)1.6 s

Accepted Answer

The answer of A thin, circular hoop with a radius...

Question 20

Complete the following sentence: Resonance occurs in harmonic motion when
A)the system is overdamped.
B)the system is critically damped.
C)the energy in the system is a minimum.
D)the driving frequency is the same as the natural frequency of the system.
E)the energy in the system is proportional to the square of the motion's amplitude.

Accepted Answer

Resonance occurs in harmonic motion when the driving frequency is the same as the natural frequency of the system. This causes the amplitude of the motion to increase significantly, as energy is transferred efficiently from the driving source to the system. This phenomenon is important in many areas of physics, including acoustics, electronics, and structural engineering.

Quiz 10: Simple Harmonic Motion and Elasticity

The spring constant for the spring in a special cannon is 1800 N/m. In cocking the cannon, the spring is compressed 0.55 m. What is the initial speed of a 7.0-kg cannonball at rest on the free end of the spring when it is released?

In a certain clock, a pendulum of length L1 has a period T1 = 0.95 s. The length of the pendulum is adjusted to a new value L2 such that T2 = 1.0 s. What is the ratio L2/L1?

Which one of the following terms is used to describe a system in which the degree of damping is just enough to stop the system from oscillating?

A relaxed spring protrudes from an opening 0.0500 meters as shown in Figure A. A 1.00-kg block is found to just force the spring completely into the opening as shown in Figure B. -How much potential energy is stored in the spring in Figure B?

A simple harmonic oscillator with a period of 2.0 s is subject to damping so that it loses one percent of its amplitude per cycle. About how much energy does this oscillator lose per cycle?

A simple pendulum on earth has a period of 6.0 s. What is the approximate period of this pendulum on the moon where the acceleration due to gravity is roughly 1/6 that of earth?

Complete the following statement: Young's modulus cannot be applied to

What are the SI units of the shear modulus?

A relaxed spring protrudes from an opening 0.0500 meters as shown in Figure A. A 1.00-kg block is found to just force the spring completely into the opening as shown in Figure B. -Determine the spring constant k.

What is the period of a pendulum consisting of a 6-kg object oscillating on a 4-m string?

A cable stretches by an amount d as it supports a crate of mass M. The cable is cut in half. What is the mass of the load that can be supported by either half of the cable if the cable stretches by an amount d?

A cable stretches by an amount d when it supports a crate of mass M. The cable is replaced by another cable of the same material having the same length and twice the diameter. What is the mass of the load that can be supported by the thicker cable if it stretches by an amount d?

A cable stretches by an amount d when it supports a crate of mass M. The cable is replaced by another cable of the same material having the same length and twice the diameter. If the same crate is supported by the thicker cable, by how much will the cable stretch?

A cable stretches by an amount d when it supports a crate of mass M. The cable is then cut into three equal pieces. If the same crate is supported by any one of these pieces, by how much will the cable stretch?

A pendulum is transported from sea-level, where the acceleration due to gravity g is 9.80 m/s2, to the bottom of Death Valley. At this location, the period of the pendulum is decreased by 3.00%. What is the value of g in Death Valley?

A thin, circular hoop with a radius of 0.32 m is hanging on a nail. Adam notices that the hoop is oscillating back and forth through small angles like a physical pendulum. The moment of inertia of the hoop for the rotational axis passing through the nail is I = 2mr2. What is the period of the hoop?

Complete the following sentence: Resonance occurs in harmonic motion when

In a certain clock, a pendulum of length L₁ has a period T₁ = 0.95 s. The length of the pendulum is adjusted to a new value L₂ such that T₂ = 1.0 s. What is the ratio L₂/L₁?

A pendulum is transported from sea-level, where the acceleration due to gravity g is 9.80 m/s², to the bottom of Death Valley. At this location, the period of the pendulum is decreased by 3.00%. What is the value of g in Death Valley?