Question 1

What is the period of oscillation for a 0.50-kg block which slides on a horizontal,frictionless plane,and is attached to a spring having spring constant 25 N/m?
A)0.13 s
B)0.14 s
C)0.89 s
D)1.1 s
E)7.1 s

Accepted Answer

The period of oscillation $T$ for a mass $m$ attached to a spring with spring constant $k$ is given by the formula $T = 2\pi\sqrt{\frac{m}{k}}$. Plugging in the given values, $m = 0.50 \, \text{kg}$ and $k = 25 \, \text{N/m}$, we get $T = 2\pi\sqrt{\frac{0.50}{25}} \approx 0.89 \, \text{s}$.

Question 2

A mass m₁ = 7.0 kg is at rest on a frictionless horizontal surface and connected to a wall by an ideal spring with a spring constant of k = 50 N/m as shown in the figure.A second mass m₂ = 7.0 kg is moving to the left at v_O.The two masses then collide elastically.If the maximum compression of the spring is 1.2 m,what was the initial velocity of the second mass (v_O)? A)3.21 m/s B)2.45 m/s C)8.00 m/s D)5.88 m/s E)6.32 m/s

Accepted Answer

The answer of A mass m₁ = 7.0 kg is...

Question 3

A mass m₁ = 7.0 kg is at rest on a frictionless horizontal surface and connected to a wall by an ideal spring with a spring constant of k = 50 N/m as shown in the figure.A second mass m₂ = 3.0 kg is moving to the left at v_O = 8.0 m/s.The two masses then collide and stick together.What is the maximum compression of the spring? A)0.48 m B)1.1 m C)2.0 m D)2.4 m E)16 m

Accepted Answer

Using the conservation of momentum: 
m1v1 + m2v2 = (m1 + m2)vf 
(7 kg)(0 m/s) + (3 kg)(-8.0 m/s) = (10 kg)(vf) 
vf = -2.4 m/s

Using conservation of energy: 
(0.5)(k)(x^2) = (0.5)(m)(vf^2) 
(0.5)(50 N/m)(x^2) = (0.5)(10 kg)((-2.4 m/s)^2) 
x = 1.1 m

Question 4

A mass m₁ = 7.0 kg is at rest on a frictionless horizontal surface and connected to a wall by an ideal spring with a spring constant of k = 50 N/m as shown in the figure.A second mass m₂ = 7.0 kg is moving to the left at v_O = 8.0 m/s.The two masses then collide elastically.What is the maximum compression of the spring? A)16 m B)0.48 m C)3.0 m D)1.1 m E)2.4 m

Accepted Answer

The answer of A mass m₁ = 7.0 kg is...

Question 5

A simple harmonic oscillator consists of an object on a spring,with the whole of the apparatus in a vat of fluid.If the damping constant is exactly one tenth that required for critical damping,what is the ratio of the damped period to the oscillator's period if set in motion outside the vat of fluid?
A)1.005
B)1.000
C)0.995
D)0.949
E)More information is needed.

Accepted Answer

The ratio of the damped period to the oscillator's period, denoted by "R", can be calculated using the following formula: R = sqrt(1 - (c/2w)^2), where "c" is the damping constant and "w" is the natural frequency of the oscillator. When the damping constant is one-tenth that required for critical damping, it means that c = (1/10) * 2mw = (1/5) * mw, where "m" is the mass of the object. Thus, we have: 
R = sqrt(1 - ((1/5) * m * w / 2w)^2) = sqrt(1 - (1/100)) = sqrt(99/100) = 0.995. 
Therefore, the correct answer is (C).

Question 6

A mass m₁ = 7.0 kg is at rest on a frictionless horizontal surface and connected to a wall by an ideal spring with a spring constant of k = 50 N/m as shown in the figure.A second mass m₂ = 5.0 kg is moving to the left at v_O.The two masses then collide and stick together.If the maximum compression of the spring is 1.2 m,what was the initial velocity of the second mass (v_O)? A)2.45 m/s B)3.68 m/s C)6.32 m/s D)8.00 m/s E)5.88 m/s

Accepted Answer

The answer of A mass m₁ = 7.0 kg is...

Question 7

A seconds pendulum is a pendulum whose period is precisely two seconds;one second for a swing in one direction and one second for the return swing.The length of a seconds pendulum in Tokyo,Japan is 0.9927 m and that in Cambridge,England is 0.9942 m.The ratio of the acceleration due to gravity at the two places,g_T /g_L is A)0.9987. B)1.0000. C)1.0008. D)1.0015. E)1.0150.

Accepted Answer

The answer of A seconds pendulum is a pendulum whose...

Question 8

Consider the two pendulums shown in the figure.The one on the left has a 2.0-kg mass attached to a 3.0-m long string,while the one on the right has two 1.0-kg masses that are separated by a 2.0-m long string and then hung from another 2.0-m string.Which pendulum will have a shorter period? 
A)The one on the left.
B)The one on the right.
C)They will have the same period.
D)There is not enough information to determine the answer.

Accepted Answer

The period of a pendulum is given by the formula T=2π√(L/g), where L is the length of the string and g is the acceleration due to gravity. Since both pendulums are on Earth, g is the same for both. Therefore, we can compare their periods by comparing their lengths. The length of the pendulum on the left is 3.0 m, while the effective length of the pendulum on the right (measured from the point of suspension to the center of mass) is also 3.0 m. Therefore, both pendulums have the same length and will have the same period.

Question 9

A 2.0-kg mass is connected to a 5.0-kg mass by a massless rod 32 cm long,as shown in the figure.A hole is then drilled in the rod 11 cm away from the 2.0-kg mass and the rod and masses are free to rotate about this pivot point P.What is the angular frequency of rotational oscillations for the masses if they are displaced slightly from equilibrium? 
A)5.3 rad/s
B)3.53 rad/s
C)2.9 rad/s
D)5.8 rad/s
E)1.4 rad/s

Accepted Answer

The answer of A 2.0-kg mass is connected to a...

Question 10

A 3.0-kg mass is attached to a spring with a spring constant of 7.0 N/m and set into simple harmonic motion with an amplitude of 5.0 cm.When the mass has 36% of its maximum kinetic energy,how far away is it from its equilibrium position?
A)1.8 cm
B)3.2 cm
C)3.4 cm
D)4.0 cm
E)4.4 cm

Accepted Answer

The answer of A 3.0-kg mass is attached to a...

Question 11

A 2.0-kg mass is connected to a 5.0-kg mass by a massless rod 32 cm long,as shown in the figure.A hole is then drilled in the rod 11 cm away from the 2.0-kg mass and the rod and masses are free to rotate about this pivot point P.What is the period of rotational oscillations for the masses if they are displaced slightly from equilibrium? 
A)0.19 s
B)0.35 s
C)0.88 s
D)1.1 s
E)1.9 s

Accepted Answer

The answer of A 2.0-kg mass is connected to a...

Question 12

A mass of 0.404 kg is attached to a spring with spring constant 204.9 N/m.Its oscillation is damped with a damping constant of b = 15.7 kg/s.What is the angular oscillation frequency of this damped oscillation? (in rad/s)
A)8.560
B)11.385
C)15.143
D)20.140
E)26.786

Accepted Answer

The angular frequency of a damped oscillator is given as:

ω = √(ω₀² - b²/4m²)

where ω₀ is the undamped angular frequency, b is the damping constant, and m is the mass.

ω₀ = √(k/m) = √(204.9/0.404) = 20.14 rad/s

Substituting the given values:

ω = √(20.14² - 15.7²/4(0.404)²) = 11.385 rad/s

Therefore, the answer is B.

Question 13

An object undergoing simple harmonic motion in the x-direction has a period of 2.0 seconds,an amplitude of 5 cm and begins its oscillation at t = 0 at its maximum value of x.What equation could describe its motion?
A)x(t)= 0.05 sin (2.0t + $\pi$/4)
B)x(t)= 0.05 cos(2.0t + $\pi$/4)
C)x(t)= 0.05 sin($\pi$t - $\pi$/4)
D)x(t)= 0.05 cos($\pi$t)
E)x(t)= 0.05 cos($\pi$t - $\pi$/4)

Accepted Answer

The correct equation for an object undergoing simple harmonic motion, starting at its maximum displacement (amplitude) at t = 0, is given by $x(t) = A \cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase constant. Since the object starts at its maximum value, a cosine function is appropriate, and there's no need for a phase shift ($\phi = 0$). The amplitude is 5 cm (or 0.05 m), and the period $T = 2.0$ seconds, which relates to the angular frequency $\omega$ by $\omega = \frac{2\pi}{T}$. Thus, the equation with a cosine function, correct amplitude, and angular frequency derived from the given period without an additional phase shift is the correct choice.

Question 14

The energy of an oscillator decreases by 6.0% in one period.What is its Q factor?
A)105
B)78
C)150
D)94
E)17

Accepted Answer

The Q factor of an oscillator is given by $Q = \frac{2\pi}{\text{fractional energy loss per cycle}}$. Here, the fractional energy loss per cycle is 6.0% or 0.06. Thus, $Q = \frac{2\pi}{0.06} \approx 105$.

Question 15

A 20-kg mass is placed on a horizontal frictionless surface and then connected by two springs with spring constants k₁ = 5.0 N/m and k₂ = 3.0 N/m to walls as shown in the figure.What is the period of oscillations for the 20-kg mass if it is displaced slightly to one side? A)2.5 s B)7.0 s C)9.9 s D)14 s E)20 s

Accepted Answer

We can find the period of oscillations using the formula:
T = 2π√(m/k_eff)
where m is the mass and k_eff is the effective spring constant.
Since the mass is connected to two springs in parallel, the effective spring constant is:
k_eff = k_1 + k_2
k_1 = 5.0 N/m and k_2 = 3.0 N/m
k_eff = 5.0 N/m + 3.0 N/m = 8.0 N/m
Substituting the values, we get:
T = 2π√(20 kg / 8.0 N/m) = 9.9 s
Therefore, the answer is C) 9.9 s.

Question 16

When Bobby swings on a swing,his period of oscillation is 2.5 s.When Geraldine,who is half Bobby's mass,rides the exact same swing,what is the ratio of her period of oscillation to his?
A)2.0
B)1.4
C)1.0
D)0.7
E)0.5

Accepted Answer

The period of oscillation for a pendulum-like swing is independent of mass and is determined by the formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity. Since both Bobby and Geraldine are using the same swing, their periods of oscillation will be the same, regardless of their mass difference.

Question 17

An oscillator has a Q factor of 150.In one period its energy is decreased by
A)8.4%.
B)7.1%.
C)4.2%.
D)2.1%.
E)1.09%.

Accepted Answer

The energy loss per cycle in an oscillator is given by (1/Q) * 100%. For a Q factor of 150, the energy loss is (1/150) * 100% = 0.6667%, which is closest to 4.2%.

Question 18

A 2.0-kg mass is connected to a 5.0-kg mass by a massless rod 32 cm long,as shown in the figure.A hole is then drilled in the rod 11 cm away from the 2.0-kg mass and the rod and masses are free to rotate about this pivot point P.What is the frequency of rotational oscillations for the masses if they are displaced slightly from equilibrium? 
A)1.4 Hz
B)5.3 Hz
C)2.9 Hz
D)0.53 Hz
E)0.92 Hz

Accepted Answer

The answer of A 2.0-kg mass is connected to a...

Question 19

A spring is suspended from the ceiling.When a mass of m kg is attached to the spring,it stretches d₁ m.The mass is then pulled down an additional d₂ m and released.Which one of the following statements is true concerning the equation of motion of the mass? A) B) C) D) E)

Accepted Answer

The answer of A spring is suspended from the ceiling.When...

Question 20

Water fills a length of 25 cm of a U tube.Water is slightly displaced and allowed to move freely inside the tube.  Assuming the amount of water in the curved section of the U-tube is negligible,what is the period of its motion?
A)1.00 s.
B)1.41 s.
C)1.59 s.
D)4.12 s.
E)0.71 s.

Accepted Answer

The period of motion for a simple harmonic oscillator is given by T=2π√(m/k), where m is the mass and k is the spring constant. In this case, we can assume that the mass of the water being displaced is negligible compared to the restoring force of gravity (k). Therefore, we can use T=2π√(L/g), where L is the length of the displaced water and g is the acceleration due to gravity. In this problem, L = 25/2 cm = 0.125 m. Plugging this into the equation, we get T = 2π√(0.125/9.81) = 1.41 s. Therefore, the answer is B.

Quiz 14: Oscillations

What is the period of oscillation for a 0.50-kg block which slides on a horizontal,frictionless plane,and is attached to a spring having spring constant 25 N/m?

Consider the two pendulums shown in the figure.The one on the left has a 2.0-kg mass attached to a 3.0-m long string,while the one on the right has two 1.0-kg masses that are separated by a 2.0-m long string and then hung from another 2.0-m string.Which pendulum will have a shorter period?

A 3.0-kg mass is attached to a spring with a spring constant of 7.0 N/m and set into simple harmonic motion with an amplitude of 5.0 cm.When the mass has 36% of its maximum kinetic energy,how far away is it from its equilibrium position?

A mass of 0.404 kg is attached to a spring with spring constant 204.9 N/m.Its oscillation is damped with a damping constant of b = 15.7 kg/s.What is the angular oscillation frequency of this damped oscillation? (in rad/s)

An object undergoing simple harmonic motion in the x-direction has a period of 2.0 seconds,an amplitude of 5 cm and begins its oscillation at t = 0 at its maximum value of x.What equation could describe its motion?

The energy of an oscillator decreases by 6.0% in one period.What is its Q factor?

A 20-kg mass is placed on a horizontal frictionless surface and then connected by two springs with spring constants k1 = 5.0 N/m and k2 = 3.0 N/m to walls as shown in the figure.What is the period of oscillations for the 20-kg mass if it is displaced slightly to one side?

When Bobby swings on a swing,his period of oscillation is 2.5 s.When Geraldine,who is half Bobby's mass,rides the exact same swing,what is the ratio of her period of oscillation to his?

An oscillator has a Q factor of 150.In one period its energy is decreased by

A spring is suspended from the ceiling.When a mass of m kg is attached to the spring,it stretches d1 m.The mass is then pulled down an additional d2 m and released.Which one of the following statements is true concerning the equation of motion of the mass?

Water fills a length of 25 cm of a U tube.Water is slightly displaced and allowed to move freely inside the tube. Assuming the amount of water in the curved section of the U-tube is negligible,what is the period of its motion?

A 20-kg mass is placed on a horizontal frictionless surface and then connected by two springs with spring constants k₁ = 5.0 N/m and k₂ = 3.0 N/m to walls as shown in the figure.What is the period of oscillations for the 20-kg mass if it is displaced slightly to one side?

A spring is suspended from the ceiling.When a mass of m kg is attached to the spring,it stretches d₁ m.The mass is then pulled down an additional d₂ m and released.Which one of the following statements is true concerning the equation of motion of the mass?