Simple Harmonic Motion

Simple Harmonic Motion

Quantities that are a maximum at (a) and (c)

• Displacement

• Force ( F = kx)

• Acceleration ( a = F/m)

• Elastic Potential Energy ( U_s = (1/2)kx²)

Quantities that are zero at (a) and (c)

• Velocity

• Kinetic Energy ( K = (1/2)mv²)

• Momentum ( p = mv)

Quantities that are a zero at (b)

• Displacement

• Force

• Acceleration

• Elastic Potential Energy

Quantities that are a maximum at (b)

• Velocity

• Kinetic Energy

• Momentum

A pendulum closely follows SHM

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• l = length of pendulum

Multiple Choice

A simple pendulum has a period of 2 s for small amplitude oscillations.

The length of the pendulum is most nearly

• 1/6 m

• 1/4 m

• 1/2 m

• 1 m

• 2 m

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• k = spring constant

Multiple Choice

A 0. l -kilogram block is attached to an initially unstretched spring of force constant k = 40 newtons per meter as shown above. The block is released from rest at time t = 0.

At what time after release will the block first return to its initial position?

• $$\frac{\pi}{40}\ s$$

• $$\frac{\pi}{20}\ s$$

• $$\frac{\pi}{10}\ s$$

• $$\frac{\pi}{5}\ s$$

• $$\frac{\pi}{4}\ s$$

Multiple Choice

An object swings on the end of a cord as a simple pendulum with period T. Another object oscillates up and down on the end of a vertical spring. also with period T. If the masses of both objects are doubled. what are the new values for the Pendulum Period, TP, and the Spring Period, Ts?

• $$T_P\ =\ \frac{T_{ }}{\sqrt{2}}$$  and  $$T_S\ =\ T_{ }\sqrt{2}$$  

• $$T_P\ =\ T$$   and  $$T_S\ =\ T_{ }\sqrt{2}$$  

• $$T_P\ =\ T$$  and  $$T_S\ =\ T$$  

• $$T_P\ =\ T\sqrt{2}$$  and  $$T_P\ =\ T$$  

If we "traced" the position of the mass over time we would get

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  A sine (or cosine) wave

Multiple Choice

An object is attached to a spring and oscillates with amplitude A and period T, as represented on the graph above. The nature of the velocity v and acceleration a of the object at time T/4 is best represented by which of the following?

• v > 0, a > 0

• v > 0, a < 0

• v > 0, a = 0

• v = 0, a < 0

• v = 0, a = 0

Fill in the Blanks

What is the amplitude of this motion?

Fill in the Blanks

What is the period of this motion? (just type the number)

Multiple Choice

The frequency of the motion would be closest to..

• 0.125 Hz

• 0.16 Hz

• 2π*6 Hz

• 8 Hz

Multiple Choice

Which equation best describes the position of the mass as a function of time?

• x = 2sin(2π(0.16)t)

• x = 2cos(2π(0.16)t)

• x = 0.16sin(2π(2)t)

• x = 0.16cos(2π(2)t)

• velocity vs t graphs

• and acceleration vs time graphs

• Remember...

• derivative of cos = -sin

• derivative of sin = cos

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​From the x vs t graph we can get

Multiple Choice

A mass is hung from an ideal spring.  It is pulled down a distance y in the negative direction and released.  Which shows the correct position vs. time and corresponding velocity vs. time graphs.

• A

• B

• C

• D

A graph of ENERGY vs position gives

• Notice PE is a max at A

• KE is a max in the middle

• Total Energy is constant

A graph of energy vs TIME gives

Multiple Choice

Which of the following is true for a system consisting of a mass oscillating on the end of an ideal spring?

• The kinetic and potential energies are equal at all times.

• The kinetic and potential energies are both constant.

• The maximum potential energy is achieved when the mass passes through its equilibrium position.

• The maximum kinetic energy and maximum potential energy are equal, but occur at different times.

• The maximum kinetic energy occurs at maximum displacement of the mass from its equilibrium position.

Multiple Choice

A 1.0 kg mass is attached to the end of a vertical ideal spring with a force constant of 400 N/m. The mass is set in simple harmonic motion with an amplitude of 10 cm. The speed of the 1.0 kg mass at the equilibrium position is

• 2 m/s

• 4 m/s

• 20 m/s

• 40 m/s

• 200 m/s