Oscillation

A mass is connected to the end of a spring and undergoes simple harmonic oscillation. The graph provided shows the position of the mass as measured from the floor as a function of time.

8. What is the period of the mass's oscillation (in units of seconds)?

(A) 2.0
(B) 4.0
(C) 5.0
(D) 8.0
(E) 10.0

9. What is the amplitude of the mass's oscillation (in units of meters)?

(A) 1.0
(B) 2.0
(C) 3.0
(D) 4.0
(E) 5.0

A point object of mass $M$ is connected to the end of a long string of negligible mass and the system swings as a simple pendulum with period $T$. The point object of mass $M$ is now replaced with a point object of mass $4M$. When this new system swings as a simple pendulum, what is its period?

(A) 4 $T$
(B) 2 $T$
(C) $T$
(D) $T/2$
(E) $T/4$

A simple pendulum of length $L$ has a point mass $M$ released from rest from the horizontal position shown. In the absence of air resistance and friction, the mass swings through the arc of a circle. Let $T$ represent the magnitude of the force from the string on the mass (tension), $G$ represent the magnitude of the gravitational force acting on the mass by the earth, and $F$ represent the magnitude of the net force acting on the mass. Which one of the following choices describes the relationship among these forces when the mass swings at the bottom of the arc (point P)?

(A) G (B) F (C) F (D) T (E) G=F

A simple pendulum oscillates with a period of 2.0 $s$. If the maximum oscillation of the pendulum is $4.0^{\circ}$ from equilibrium, what is the length of the string for this pendulum?

(A) 6.4 $m$
(B) 3.2 $m$
(C) 1.6 $m$
(D) 1.0 $m$
(E) 0.5 $m$

A simple pendulum consists of a massive bob connected to the end of a very light string. Which one of the following changes should be made in order to increase the period of the pendulum? Ignore air resistance.

(A) Increase the mass of the bob

(B) Decrease the mass of the bob

(C) Increase the length of the string

(D) Decrease the length of the string

(E) Decrease the maximum angle of the pendulum's oscillation

A 2.50 $kg$ mass connected to the end of an ideal spring oscillates in simple harmonic motion. The mass's position is described as a function of time by $x(T)=0.20\cos(8.00T+0.50)$ where all quantities are in base SI units. Which one of the following choices gives the numerical value of the oscillation's amplitude in base SI units?

(A) 8.00
(B) 4.00
(C) 1.60
(D) 0.50
(E) 0.20

A long thin rod of mass $M$ and length $L$ is pivoted at one end so that it swings as a pendulum. The rod is set into simple harmonic oscillation and has a period of motion $T$. A second thin rod with mass $2M$ and length $2L$ also is pivoted at one end to swing as a pendulum. When this second rod is set into simple harmonic oscillation, what is its period?

(A) 2 $T$
(B) $\sqrt{2}T$
(C) $T$
(D) $\frac{1}{\sqrt{2}}T$
(E) $\frac{1}{2}T$

The position of a mass connected to a spring obeys $x(\mathrm{t})=A\cos(\omega \mathrm{t})$ . What is the average speed of the mass for one full oscillation in terms of the mass's maximum speed during oscillation, $v_{\max}$?

(A) $\frac{2}{\pi}v_{\max}$
(B) $\frac{1}{\sqrt{2}}v_{\max}$
(C) $\frac{1}{2}v_{\max}$
(D) $\frac{\sqrt{2}}{\pi}v_{\max}$
(E) $\frac{1}{2\pi\sqrt{2}}v_{\max}$

A mass connected to a string forms a simple pendulum. The mass is released from rest and undergoes simple harmonic motion. Which one of the following choices correctly ranks the magnitude of the tension in the string (T) , the gravitational force on the mass (W) , and the net force acting on the mass (F) when the mass swings through its lowest point?

(A) $F<W=T$

(B) $W<T<F$

(C) $W<F=T$

(D) $F<W<T$

(E) More information is required to answer the question.

A mass connected to an ideal spring oscillates with a period $T$ when it is released from rest at a position $A$ from equilibrium. Which one of the following choices correctly identifies the period of oscillation for the mass connected to the same spring when it is released from rest at a position $2A$ from equilibrium?

(A) $\frac{1}{4}T$
(B) $\frac{1}{2}T$
(C) $T$
(D) $2T$
(E) $4T$

A simple pendulum consists of a mass $M$ attached to a string of length $L$ of negligible mass. For this system, when undergoing small oscillations

(A) the frequency is proportional to the amplitude.

(B) the period is proportional to the amplitude.

(C) the frequency is independent of the mass $M.$

(D) the frequency is independent of the length $L.$

(E) the frequency is inversely proportional to the length $L$

A 0.30 kg mass is suspended on a vertical spring. In equilibrium the mass stretches the spring 2.0 cm downward. The mass is then pulled an additional distance of l.0 cm down and released from rest. What is the period of oscillation?

(A) 0.14 $\mathrm{s}$
(B) 0.28 $\mathrm{s}$
(C) 0.24 $\mathrm{s}$
(D) 0.46 $\mathrm{s}$
(E) 0.64 $\mathrm{s}$

A mass that is in simple harmonic motion obeys the following position versus time equation: $\mathrm{y}=(\mathrm{0}.50\mathrm{m})\sin(\pi/2\mathrm{t})$ where $\mathrm{t}$ is in seconds. What is the period of vibration of this mass?

(A) 1.0 $\mathrm{s}$
(B) 2.0 $\mathrm{s}$
(C) 3.0 $\mathrm{s}$
(D) 4.0 $\mathrm{s}$
(E) 5.0 $\mathrm{s}$

A person places some water on their fingertip and rubs the fingertip around the top surface of a water glass to produce a particular musical tone. This process is most closely associated with which of the following phenomena?
(A) Diffraction
(B) Dispersion
(C) Doppler effect
(D) Refraction
(E) Resonance

Two separate simple pendulums are swinging back and forth. It is found that pendulum A has twice the frequency of pendulum $B$. Which statement could account for this difference?
(A) Pendulum $B$ is twice as long as pendulum $A$.
(B) Pendulum $B$ was released before pendulum A .
(C) Pendulum $B$ is twice as massive as pendulum A .
(D) The length of pendulum $B$ is four times the length of pendulum $A$.
(E) The mass of pendulum $B$ is four times the mass of pendulum $A$.

Two identical uniform sticks have small holes drilled in them. The sticks are marked with equally spaced lines as shown. Each stick is suspended from a pin through the hole and is free to rotate about the pin. Both sticks are displaced a small angle from the vertical and released. What is the ratio of the period of pendulum one to the period of pendulum two?

(A) $1: 1$
(B) $2: 3$
(C) $4: 3$
(D) $10: 9$
(E) $16: 9$

A 4.0 kg mass at the end of a spring moves with simple harmonic motion on a horizontal frictionless table with a period of 2.0 s and an amplitude of 2.0 m. Determine the maximum force exerted on the spring.
(A) 25.1 N
(B) 158 N
(C) 39.5 N
(D) 63.0 N
(E) 79.0 N

Two simple pendula are 60 cm and 63 cm in length. They hang vertically, one in front of the other. If they are set in motion simultaneously, find the time taken for one to gain a complete cycle of oscillation on the other.
(A) 15.4 $\mathrm{~s}$
(B) 15.7 $\mathrm{~s}$
(C) 31.5 $\mathrm{~s}$
(D) 62.4 $\mathrm{~s}$
(E) 79.7 $\mathrm{~s}$

A 0.200 kg mass attached to the end of a spring moves up and down through 10 cycles in 6.50 s . What is the force constant of the spring
(A) $18.7 \mathrm{~N} / \mathrm{m}$
(B) $4.67 \mathrm{~N} / \mathrm{m}$
(C) $22.8 \mathrm{~N} / \mathrm{m}$
(D) $5.95 \mathrm{~N} / \mathrm{m}$
(E) 12.0 N

A pendulum clock has a period of 1.0 s and gives the correct time on the ground at a certain location. It is then moved to the top of a building that is 320 m high. How much time will the pendulum lose in 1 day at this height?
(A) 2.25 s
(B) 2.88 s
(C) 3.42 s
(D) 3.94 s
(E) 4.32 s

Atoms in a solid are not stationary. They vibrate about their equilibrium positions. Typically, the frequency of vibration is about $2.0 \times 10^{12} \mathrm{~Hz}$, and the amplitude is about $1.1 \times 10^{-11} \mathrm{~m}$. For a typical atom, what is its maximum speed?
(A) $108 \mathrm{~m} / \mathrm{s}$
(B) $10.8 \mathrm{~m} / \mathrm{s}$
(C) $128 \mathrm{~m} / \mathrm{s}$
(D) $12.8 \mathrm{~m} / \mathrm{s}$
(E) $138 \mathrm{~m} / \mathrm{s}$

A pendulum has a length of 2.00 m and has a mass attached to its end. The mass is pulled back $60.0^{\circ}$ from the vertical before being released and allowed to swing downward. If the string breaks under a tension of more than 20 N , what is the maximum amount of mass that can be attached to string to allow the mass to swing without breaking the string?
(A) 0.8 kg
(B) 1.0 kg
(C) 1.2 kg
(D) 1.4 kg
(E) 1.6 kg

A pendulum is constructed from a uniform rod pivoted at its top end and a circular disk attached near the free end as shown in Figure 1. Figure 2 shows the same pendulum with the disk now attached closer to the pivot. Use the following statements to determine the resulting change in the period of the pendulum.

Statement 1: The moment of inertia of the pendulum about the pivot decreases.
Statement 2: The torque exerted by gravity on the pendulum about the pivot decreases.
(A) Only statement one is true, resulting in a decreased period of the pendulum.
(B) Only statement two is true, resulting in an increased period of the pendulum.
(C) Both statements are true, resulting in a decreased period of the pendulum.
(D) Both statements are true, resulting in no change in the period of the pendulum.
(E) Both statements are true, resulting in an increased period of the pendulum.

A $2.0-\mathrm{kg}$ mass suspended from a vertical spring stretches it a distance of 5.0 cm . The mass is then pulled down a distance of 10 cm and released. What is the frequency of oscillation?
(A) 0.113 Hz
(B) 2.25 Hz
(C) 3.11 Hz
(D) 0.159 Hz
(E) 1.59 Hz

A mass on a spring executes simple harmonic motion with an amplitude of 10.0 cm and a period of 2.0 s . The average velocity of the mass over one complete oscillation is ...
(A) $\quad 0.0 \frac{\mathrm{~cm}}{\mathrm{~s}}$
(B) $5.0 \frac{\mathrm{~cm}}{\mathrm{~s}}$
(C) $10.0 \frac{\mathrm{~cm}}{\mathrm{~s}}$
(D) $15.0 \frac{\mathrm{~cm}}{\mathrm{~s}}$
(E) $20.0 \frac{\mathrm{~cm}}{\mathrm{~s}}$