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### AP Physics C: Mechanics

Oscillation in-class examples

Example 1 – Spring SHM

A 12 cm-long spring has a force constant (k) of 400 N/m. How much force is required to stretch the spring to a length of 14 cm?

Example 1 – Spring SHM

A 12 cm-long spring has a force constant (k) of 400 N/m. How much force is required to stretch the spring to a length of 14 cm?

8 N

Example 2– Spring SHM

A block of mass m = 0.05 kg oscillated on a spring whose force constant k is 500 N/m. The amplitude of the oscillations is 4.0 cm. Calculate the maximum speed of the block.

Example 2– Spring SHM

A block of mass m = 0.05 kg oscillated on a spring whose force constant k is 500 N/m. The amplitude of the oscillations is 4.0 cm. Calculate the maximum speed of the block.

4 m/s

Example 3– Spring SHM

Show why Us = 1/2kx2 if Fs = -kx.

Example 4– Spring SHM

A block of mass m = 2.0 kg is attached to an ideal spring of force constant k = 500 N/m. The amplitude of the resulting oscillations is 8.0 cm. Determine the total energy of the oscillator and the speed of the block when it’s 4.0 cm from equilibrium.

Example 4– Spring SHM

A block of mass m = 2.0 kg is attached to an ideal spring of force constant k = 500 N/m. The amplitude of the resulting oscillations is 8.0 cm. Determine the total energy of the oscillator and the speed of the block when it’s 4.0 cm from equilibrium.

1.1 m/s

Example 5– Spring SHM

A block of mass m = 3.0 kg is attached to an ideal spring of force constant k = 500 N/m. The block is at rest at its equilibrium position. An impulsive force acts on the block, giving it an initial speed of 2.0 m/s. Find the amplitude of the resulting oscillations.

Example 5– Spring SHM

A block of mass m = 3.0 kg is attached to an ideal spring of force constant k = 500 N/m. The block is at rest at its equilibrium position. An impulsive force acts on the block, giving it an initial speed of 2.0 m/s. Find the amplitude of the resulting oscillations.

0.15 m

Example 6– Spring SHM

A block oscillating on the end of a spring moves from its position of maximum spring stretch to maximum spring compression in 0.25 s. Determine the period and frequency of this motion.

Example 6– Spring SHM

A block oscillating on the end of a spring moves from its position of maximum spring stretch to maximum spring compression in 0.25 s. Determine the period and frequency of this motion.

T = 0.5 s, f = 2 Hz

Example 7– Spring SHM

A student observing an oscillating block counts 45.5 cycles of oscillation in one minute. Determine its frequency (in hertz) and period (in seconds).

Example 7– Spring SHM

A student observing an oscillating block counts 45.5 cycles of oscillation in one minute. Determine its frequency (in hertz) and period (in seconds).

f = 0.758 Hz

T = 1.32 s

Example 8– Spring SHM

A block of mass m = 2.0 kg is attached to a spring whose force constant, k, is 300 N/m. Calculate the frequency and period of the oscillations of this spring-block system.

Example 8– Spring SHM

A block of mass m = 2.0 kg is attached to a spring whose force constant, k, is 300 N/m. Calculate the frequency and period of the oscillations of this spring-block system.

f = 1.9 Hz

T = 0.51 s

Example 9– Spring SHM

A block is attached to a spring and set into oscillatory motion, and its frequency is measured. If this block were removed and replaced by a second block with ¼ the mass of the first block, how would the frequency of the oscillations compare to that of the first block?

Example 9– Spring SHM

A block is attached to a spring and set into oscillatory motion, and its frequency is measured. If this block were removed and replaced by a second block with ¼ the mass of the first block, how would the frequency of the oscillations compare to that of the first block?

f increases by a factor of 2.

Example 10– Spring SHM

A student performs an experiment with a spring-block simple harmonic oscillator. In the first trial, the amplitude of the oscillations is 3.0 cm, while in the second trial (using the same spring and block), the amplitude of the oscillations is 6.0 cm. Compare the values of the period, frequency, and maximum speed of the block between these two trials).

Example 10– Spring SHM

A student performs an experiment with a spring-block simple harmonic oscillator. In the first trial, the amplitude of the oscillations is 3.0 cm, while in the second trial (using the same spring and block), the amplitude of the oscillations is 6.0 cm. Compare the values of the period, frequency, and maximum speed of the block between these two trials).

T and f are the same, vmax will be twice as great

Example 11– Spring SHM

Determine the effective spring constant, keff of the three scenarios on the board in terms of k1and k2.

Example 11– Spring SHM

Determine the effective spring constant, keff of the three scenarios on the board in terms of k1and k2.

k1 + k2

k1 + k2

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