AP Review Simple Harmonic Motion

Simple Harmonic Motion Review

Mrs Wendt
AP Physics

Simple Harmonic Motion

If the displacement is directly proportional to the linear restoring force, the object undergoes simple harmonic motion.
Examples:
Mass on a spring
Pendulum

​Cycles

•Cycle: something that repeats in time at regular intervals, such as one full swing of a pendulum.

•All Harmonic Motion is a repeated sequence of cycles.

•Oscillator: system that exhibits harmonic motion.
•An orbit and a rotation are both cycles because they are repeating motions. Both are the basis for calendars.

•The Earth–Sun system has an orbital cycle of one year.

•The Earth–Moon system has a orbital cycle of approximately one month. Earth itself has several cycles. Earth rotates on its axis once a day, creating the 24-hour cycle of day and night.
•There is also a wobble of the Earth’s axis, moving the orientation of the North and South Poles around by hundreds of miles every 22,000 years.

​Vibrational Motion

•The back-and-forth motion of an object that passes through the same positions, first moving in one direction and then in the opposite direction, is an important feature of vibrational motion.

•equilibrium position—the place where it resides when not disturbed. When the cart is displaced on either side of the equilibrium position.

Restoring Force

•the spring exerts a so-called restoring force on the cart that tends to return it to that equilibrium position.

•Restoring Force: When an object is displaced from equilibrium, some other object exerts a force with a component that always points opposite the direction of the vibrating object’s displacement from equilibrium. This force tends to restore the vibrating object back toward equilibrium

​Mass on a spring

​Period of a Mass on a Spring

​Uniform circular Motion and Simple Harmonic Motion

Angular Frequency

Multiple Choice

An oscillating system is created by a releasing an object from a maximum displacement of 0.2 m. The object makes 60 complete oscillations in one minute. Determine the object's angular frequency.

• $$4\pi\ \frac{rad}{s}$$

• $$\pi\ \frac{rad}{s}$$

• $$3\pi\ \frac{rad}{s}$$

• $$2\pi\ \frac{rad}{s}$$

Multiple Choice

An oscillating system is created by a releasing an object from a maximum displacement of 0.2 m. The object makes 60 complete oscillations in one minute. Determine the object's position at time t=10s.

• 2.0 m

• 20 m

• 0.2 m

• 0.02 m

Multiple Choice

An oscillating system is created by a releasing an object from a maximum displacement of 0.2 m. The object makes 60 complete oscillations in one minute. Determine the time when the object is at position x=0.1 m.

• 1.67 s

• 2.50 s

• 0.25 s

• 0.167 s

Simple Harmonic Equations

​Inertia and Simple Harmonic

•Simple harmonic motion requires a restoring force to bring an object back to the equilibrium point.
•Restoring Force requires Inertia as that keeps an object moving through the equilibrium point.

•Inertia requires elasticity as that is the source of the restoring force.

•Elasticity results in the spring constant (k).

Period and Frequency

Amplitude

•Amplitude: maximum displacement of an oscillation from its equilibrium, or average, value.

•Amplitude is measured in units that match the oscillation.

•The key idea is that amplitude always describes the maximum displacement from equilibrium.

•Harmonic motion involves energy that oscillates among different forms. 

•Consider the mass on the spring. The system has kinetic energy because there is moving mass.

•The system also has elastic potential energy because the oscillation stretches a spring.

•The kinetic energy is largest when the mass moves through the equilibrium position. 

Multiple Select

A 5 kg block is attached to a 2000 N/m spring as shown and displaced a distance of 8 cm from its equilibrium position before being released. Determine the period of oscillation, the frequency, and the angular frequency for the block. (Pick two choices)

• 3.18 Hz

• 4.25 Hz

• 25 rad/s

• 20 rad/s

​Damping

•Damping: gradual decrease in amplitude and energy of a wave due to friction or other energy-loss mechanisms.

•All vibrational systems are subject to some dampening.
•Underdamped oscillation is when the damping force is less than the critical damping force. This results in the oscillation decaying slowly. 

•Overdamping occurs when oscillations come to a halt after a significant period of time has passed since the resistive force was applied.

•It moves towards the equilibrium point more slowly than a critically damped object.

•There are no oscillations.

•Critical damping is defined as the threshold between overdamping and underdamping.

•In the case of critical damping, the oscillator returns to the equilibrium position as quickly as possible, without oscillating, and passes it once at most.

Pendulum

​Energy of a Pendulum

Multiple Choice

When is the pendulum at maximum velocity

• At its lowest point

• At its highest point

• It moves at a constant velocity

• It is at rest

Open Ended

A pendulum of length 20 cm and a mass of 1 kg is displaced an angle of 10 degrees from the vertical. What is the maximum speed of the pendulum?

Multiple Choice

When does the mass move at maximum velocity in a mass-spring system?

• At equilibrium

• At the farthest point from the equilibrium

• At its hightest point

• At its lowest point

Multiple Choice

When is the pendulum at 0 velocity

• At its lowest point

• At its highest point

• It moves at a constant velocity

• It is at rest

Multiple Choice

What caused the Millennium Bridge to sway back and forth dramatically?

• The force of people's footsteps

• The force of the river current

• The weight of the bridge itself

• The force of the wind

Multiple Choice

A pair of trapeze performers at the circus is swinging from ropes attached to a large elevated platform. Suppose that the performers can be treated as a simple pendulum with a length of 16 m. Determine the period for one complete back and forth cycle.

• 2 sec
• 12 sec
• 8 sec
• 10 sec

Multiple Choice

A pendulum is 0.75 meters long and has a period of 4.17 seconds. The Pendulum is on an unknown planet.  What is the gravity of the Unknown Planet?

• 9.8
• 3.4
• 1.7
• Greater than 9.8

Multiple Choice

The number of cycles or oscillation made in one second is defined as

• time taken

• period

• frequency

• amplitude