Question 1

What is the frequency of a 1.30-m long simple pendulum in an elevator that is moving upward with a uniform acceleration of 1.20 m/s²?

Accepted Answer

The frequency of a simple pendulum is given by the formula: f = 1/(2π) √(g/l), where g is the acceleration due to gravity and l is the length of the pendulum. 
In this case, since the elevator is moving upward with a uniform acceleration of 1.20 m/s^2, the effective acceleration experienced by the pendulum will be g + a = 9.81 + 1.20 = 11.01 m/s^2. 
Substituting this value along with the given length of 1.30 m into the formula, we get: 
f = 1/(2π) √(11.01/1.30) ≈ 0.463 Hz 
Therefore, the best choice is B, 0.463 Hz.

Question 2

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum velocity is 25 m/s.What is its maximum acceleration?

Accepted Answer

The answer of A particle in simple harmonic motion has...

Question 3

The position of a particle moving along x-axis is given by x = 0.05 sin (15(rad/s)t + 0.5 rad)m.Its acceleration at t = 0.5 sec is

Accepted Answer

Acceleration is the second derivative of position with respect to time.
Taking derivative of the given position function two times we get, 
a = d^2x/dt^2 = -112.5sin(15t+0.5) m/s^2
Putting t=0.5 sec, we get
a = -112.5sin(8) m/s^2 = -11.13 m/s^2 (approx)
Therefore, the option B is correct.

Question 4

What is the phase difference between the position and the acceleration for a particle in simple harmonic motion?

Accepted Answer

In simple harmonic motion, acceleration is directly opposite to the displacement, resulting in a phase difference of 180 degrees.

Question 5

A 7.2-kg mass oscillates on a horizontal spring.The position of the object is given by the relation  (all units in MKS).The system is then submerged into a vat of oil with a damping constant of 1.42 kg/s.How long will it take before the amplitude of oscillations becomes 0.314 m?

Accepted Answer

The answer of A 7.2-kg mass oscillates on a horizontal...

Question 6

A cube of wood (side length 3.2 cm,mass 24 g)floats on the surface of a pond.A frog jumps off of it and it begins to oscillate in simple harmonic motion with a frequency of 5.0 Hz.What is the effective value of the spring constant of water?

Accepted Answer

The answer of A cube of wood (side length 3.2...

Question 7

A 5-kg block is attached to a horizontal spring of constant 500 N/m.It is pulled 20 cm from its equilibrium position and released.What is its angular frequency?

Accepted Answer

The angular frequency, $\omega$, of a mass-spring system is given by $\omega = \sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass. Plugging in the given values, $k = 500\, \text{N/m}$ and $m = 5\, \text{kg}$, we get $\omega = \sqrt{\frac{500}{5}} = \sqrt{100} = 10\, \text{rad/s}$.

Question 8

When the displacement of a mass on a spring is one half of the amplitude of its oscillation,what fraction of the total energy is potential energy?

Accepted Answer

The potential energy is proportional to the square of the displacement. When the displacement is one half of the amplitude, the potential energy is one fourth of the total energy.

Question 9

The figure shows mass m₁ resting on top of mass m₂ which is attached to a spring with spring constant 10 N/m,where m₁ = m₂ = 20 g.If the amplitude of the oscillation is 0.019 m.The masses are oscillating with SMH on a frictionless surface.What is the minimum coefficient of static friction between the two masses required to keep m₁ from slipping off m₂?

Accepted Answer

The answer of The figure shows mass m₁ resting on...

Question 10

A vertical spring is hanging from the ceiling with a mass attached to it.The mass is set to oscillate with SHM by pulling on it.Which of the following would increase the frequency of oscillation?

Accepted Answer

Adding a second identical spring in either configuration effectively increases the spring constant of the system, which increases the frequency of oscillation. Adding a spring in parallel (option A) or in series (option B) both result in a system that has a higher effective spring constant than the original single spring setup. Increasing the mass (option C) would actually decrease the frequency of oscillation, according to the formula for the frequency of a mass-spring system, $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, where $f$ is the frequency, $k$ is the spring constant, and $m$ is the mass.

Question 11

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum acceleration is 400 m/s².What is its maximum velocity?

Accepted Answer

We know that for a particle in simple harmonic motion, the maximum velocity is achieved when the acceleration is zero, which occurs at the equilibrium position. Therefore, we can use the equation for the maximum velocity in simple harmonic motion:

v_max = Aω

where A is the amplitude and ω is the angular frequency.

We are not given the amplitude, but we can relate it to the maximum acceleration using the equation:

a_max = Aω^2

Substituting the given values, we have:

400 = A(24)^2

Solving for A, we get:

A = 0.694 m

Finally, we can calculate the maximum velocity:

v_max = Aω = 0.694 x 24 = 16.656 ≈ 17 m/s.

Therefore, the answer is B.

Question 12

What is the period of a 1.30-m long simple pendulum in an elevator that is moving downward with a uniform acceleration of 1.20 m/s²?

Accepted Answer

The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g_{\text{eff}}}}$, where $L$ is the length of the pendulum and $g_{\text{eff}}$ is the effective acceleration due to gravity. In an elevator moving downward with acceleration $a$, the effective acceleration due to gravity becomes $g_{\text{eff}} = g - a$. Given $L = 1.30 \, \text{m}$, $g = 9.81 \, \text{m/s}^2$, and $a = 1.20 \, \text{m/s}^2$, we find $g_{\text{eff}} = 9.81 - 1.20 = 8.61 \, \text{m/s}^2$. Substituting these values into the formula gives $T = 2\pi\sqrt{\frac{1.30}{8.61}} \approx 2.44 \, \text{s}$.

Question 13

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum velocity is 25 m/s.What is the amplitude of its oscillation?

Accepted Answer

The answer of A particle in simple harmonic motion has...

Question 14

A 250-g metal cube connected to a light spring of constant 25 N/m oscillates on a horizontal,frictionless surface.The amplitude of its motion is 4 cm.The maximum velocity is

Accepted Answer

The answer of A 250-g metal cube connected to a...

Question 15

When the displacement of a mass on a spring is one half of the amplitude of its oscillation,what fraction of the total energy is kinetic energy?

Accepted Answer

The total energy of an oscillating mass on a spring is given by the equation:
Total energy = ½ kA^2
where k is the spring constant and A is the amplitude of oscillation.
When the displacement of the mass is one half of the amplitude, its displacement is A/2.
The total energy of the system at this displacement is:
Total energy = ½ k(A/2)^2
Total energy = ½ kA^2/4
Total energy = ¼ (½ kA^2)
Since the kinetic energy equals half of the total energy at any point in the motion, we can conclude that:
Kinetic energy = ¼ (½ kA^2) / 2
Kinetic energy = ¼ (½ kA^2) / 2
Kinetic energy = ¼ (¼ kA^2)
Kinetic energy = ¼ of the total energy.
Thus, the fraction of the total energy that is kinetic energy at this displacement is ¾ or choice C.

Question 16

The figure shows mass m₁ resting on top of mass m₂ which is attached to a spring with spring constant 10 N/m,where m₁ = m₂ = 20 g.The coefficient of static friction between the two masses is 0.6.The masses are oscillating with SMH on a frictionless surface.What is the maximum amplitude of oscillation without m₁ slipping off m₂?

Accepted Answer

The maximum amplitude of oscillation without m₁ slipping off m₂ is when the static friction force is at its maximum value, which is given by $f_s = \mu_s mg$, where $\mu_s$ is the coefficient of static friction, $m$ is the mass of the upper block, and $g$ is the acceleration due to gravity. The maximum static friction force is therefore $f_s = (0.6)(0.02 ext{ kg})(9.8 ext{ m/s}^2) = 0.1176 ext{ N}$. At the maximum amplitude of oscillation, the spring is stretched/compressed by the distance $d$ from its equilibrium position, so the force on the upper block is $fs$ in the direction of motion. The force of gravity on the upper block is $mg$ downward, and the force from the spring is $-kx$ upward (where $k$ is the spring constant and $x$ is the displacement from the equilibrium position, which is equivalent to $d$). Therefore, we have the following equation of motion for the upper block: $$fs - mg - kx = 0$$ At the maximum amplitude of oscillation, the velocity is momentarily zero and the acceleration is a maximum, so $fs$ is equal to the maximum static friction force as calculated above. Rearranging the equation, we find: $$d = x = (fs - mg)/k = (0.1176 ext{ N} - (0.02 ext{ kg})(9.8 ext{ m/s}^2))/10 ext{ N/m} = 0.024 ext{ m}$$ Therefore, the maximum amplitude of oscillation without the upper block slipping off is 0.024 m, which corresponds to choice (B).

Question 17

A 250-g metal cube connected to a light spring of constant 35 N/m oscillates on a horizontal,frictionless surface.The maximum velocity is 0.98 m/s.What is the amplitude of its motion?

Accepted Answer

The maximum velocity of the object occurs when all of the potential energy is converted to kinetic energy and vice versa. Using conservation of energy:

1/2 k A^2 = 1/2 mv^2

Where k is the spring constant, A is the amplitude, m is the mass of the object, and v is the maximum velocity.
Plugging in the given values, we get:

1/2 (35 N/m) A^2 = 1/2 (0.25 kg) (0.98 m/s)^2

Simplifying and solving for A, we get:

A = 0.083 m = 8.3 cm

Therefore, choice C is the best answer.

Question 18

The period of a 1.30-m long simple pendulum in an elevator that is 2.52 s.The elevator is accelerating uniformly.What is the acceleration of the elevator?

Accepted Answer

The period of a simple pendulum is given by:$$T = 2\pi \sqrt{\frac{L}{g}}$$where:- T is the period of the pendulum- L is the length of the pendulum- g is the acceleration due to gravityIn this case, the period of the pendulum is 2.52 s and the length of the pendulum is 1.30 m. We can use this information to solve for the acceleration due to gravity:$$2.52 s = 2\pi \sqrt{\frac{1.30 m}{g}}$$$$g = 9.81 m/s^2$$The acceleration of the elevator is equal to the difference between the acceleration due to gravity and the acceleration of the pendulum:$$a = g - \frac{4\pi^2 L}{T^2}$$$$a = 9.81 m/s^2 - \frac{4\pi^2 (1.30 m)}{(2.52 s)^2}$$$$a = 1.73 m/s^2$$Therefore, the acceleration of the elevator is 1.73 m/s^2 downward.

Question 19

What is the total energy of a 15-kg mass attached to a spring oscillating horizontally with an amplitude of 20 cm and an angular frequency of 32 rad/s?

Accepted Answer

The total energy of a simple harmonic oscillator is given by the formula:

E = 1/2 kA^2

where k is the spring constant, A is the amplitude of the oscillation.

We can find the spring constant k using the angular frequency:

k = mω^2

where m is the mass attached to the spring, and ω is the angular frequency.

Therefore, k = 15 kg x (32 rad/s)^2 = 15360 N/m.

Now we can substitute the values of k and A into the formula for energy:

E = 1/2 kA^2 = 1/2 x 15360 N/m x (0.2 m)^2 = 307.2 J.

Rounding this value to three significant figures, we get:

E = 307 J answer choice is (C).

Question 20

A 7.2-kg mass oscillates on a horizontal spring.The position of the object is given by the relation  (all units in MKS).What is the spring constant of the spring?

Accepted Answer

The answer of A 7.2-kg mass oscillates on a horizontal...

Quiz 14: Oscillations

What is the frequency of a 1.30-m long simple pendulum in an elevator that is moving upward with a uniform acceleration of 1.20 m/s²?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum velocity is 25 m/s.What is its maximum acceleration?

The position of a particle moving along x-axis is given by x = 0.05 sin (15(rad/s) t + 0.5 rad) m.Its acceleration at t = 0.5 sec is

What is the phase difference between the position and the acceleration for a particle in simple harmonic motion?

A 7.2-kg mass oscillates on a horizontal spring.The position of the object is given by the relation (all units in MKS) .The system is then submerged into a vat of oil with a damping constant of 1.42 kg/s.How long will it take before the amplitude of oscillations becomes 0.314 m?

A cube of wood (side length 3.2 cm,mass 24 g) floats on the surface of a pond.A frog jumps off of it and it begins to oscillate in simple harmonic motion with a frequency of 5.0 Hz.What is the effective value of the spring constant of water?

A 5-kg block is attached to a horizontal spring of constant 500 N/m.It is pulled 20 cm from its equilibrium position and released.What is its angular frequency?

When the displacement of a mass on a spring is one half of the amplitude of its oscillation,what fraction of the total energy is potential energy?

A vertical spring is hanging from the ceiling with a mass attached to it.The mass is set to oscillate with SHM by pulling on it.Which of the following would increase the frequency of oscillation?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum acceleration is 400 m/s².What is its maximum velocity?

What is the period of a 1.30-m long simple pendulum in an elevator that is moving downward with a uniform acceleration of 1.20 m/s²?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum velocity is 25 m/s.What is the amplitude of its oscillation?

A 250-g metal cube connected to a light spring of constant 25 N/m oscillates on a horizontal,frictionless surface.The amplitude of its motion is 4 cm.The maximum velocity is

When the displacement of a mass on a spring is one half of the amplitude of its oscillation,what fraction of the total energy is kinetic energy?

A 250-g metal cube connected to a light spring of constant 35 N/m oscillates on a horizontal,frictionless surface.The maximum velocity is 0.98 m/s.What is the amplitude of its motion?

The period of a 1.30-m long simple pendulum in an elevator that is 2.52 s.The elevator is accelerating uniformly.What is the acceleration of the elevator?

What is the total energy of a 15-kg mass attached to a spring oscillating horizontally with an amplitude of 20 cm and an angular frequency of 32 rad/s?

A 7.2-kg mass oscillates on a horizontal spring.The position of the object is given by the relation (all units in MKS) .What is the spring constant of the spring?

Quiz 14: Oscillations

What is the frequency of a 1.30-m long simple pendulum in an elevator that is moving upward with a uniform acceleration of 1.20 m/s2?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum velocity is 25 m/s.What is its maximum acceleration?

The position of a particle moving along x-axis is given by x = 0.05 sin (15(rad/s) t + 0.5 rad) m.Its acceleration at t = 0.5 sec is

What is the phase difference between the position and the acceleration for a particle in simple harmonic motion?

A 7.2-kg mass oscillates on a horizontal spring.The position of the object is given by the relation (all units in MKS) .The system is then submerged into a vat of oil with a damping constant of 1.42 kg/s.How long will it take before the amplitude of oscillations becomes 0.314 m?

A cube of wood (side length 3.2 cm,mass 24 g) floats on the surface of a pond.A frog jumps off of it and it begins to oscillate in simple harmonic motion with a frequency of 5.0 Hz.What is the effective value of the spring constant of water?

A 5-kg block is attached to a horizontal spring of constant 500 N/m.It is pulled 20 cm from its equilibrium position and released.What is its angular frequency?

When the displacement of a mass on a spring is one half of the amplitude of its oscillation,what fraction of the total energy is potential energy?

A vertical spring is hanging from the ceiling with a mass attached to it.The mass is set to oscillate with SHM by pulling on it.Which of the following would increase the frequency of oscillation?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum acceleration is 400 m/s2.What is its maximum velocity?

What is the period of a 1.30-m long simple pendulum in an elevator that is moving downward with a uniform acceleration of 1.20 m/s2?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum velocity is 25 m/s.What is the amplitude of its oscillation?

A 250-g metal cube connected to a light spring of constant 25 N/m oscillates on a horizontal,frictionless surface.The amplitude of its motion is 4 cm.The maximum velocity is

When the displacement of a mass on a spring is one half of the amplitude of its oscillation,what fraction of the total energy is kinetic energy?

A 250-g metal cube connected to a light spring of constant 35 N/m oscillates on a horizontal,frictionless surface.The maximum velocity is 0.98 m/s.What is the amplitude of its motion?

The period of a 1.30-m long simple pendulum in an elevator that is 2.52 s.The elevator is accelerating uniformly.What is the acceleration of the elevator?

What is the total energy of a 15-kg mass attached to a spring oscillating horizontally with an amplitude of 20 cm and an angular frequency of 32 rad/s?

A 7.2-kg mass oscillates on a horizontal spring.The position of the object is given by the relation (all units in MKS) .What is the spring constant of the spring?

What is the frequency of a 1.30-m long simple pendulum in an elevator that is moving upward with a uniform acceleration of 1.20 m/s²?

A particle in simple harmonic motion has an angular frequency of 24 rad/s.Its maximum acceleration is 400 m/s².What is its maximum velocity?

What is the period of a 1.30-m long simple pendulum in an elevator that is moving downward with a uniform acceleration of 1.20 m/s²?