Unit 7: Simple Harmonic Motion

For this unit you must:1. Predict which properties determine the motion of a simple harmonic oscillator and what the dependence of the

motion is on those properties2. Design a plan and collect data in order to ascertain the characteristics of the motion of a system undergoing

oscillatory motion caused by a restoring force3. Analyze data to identify qualitative or quantitative relationships between given values and variables (i.e., force,

displacement, acceleration, velocity, period of motion, frequency, spring constant, string length, mass) associated with objects in oscillatory motion to use that data to determine the value of an unknown

4. Construct a qualitative and or quantitative explanation of oscillatory behavior given evidence of a restoring force5. Calculate the expected behavior of a system using the object model (i.e., by ignoring changes in internal

structure) to analyze a situation. Then, when the model fails, justify the use of conservation of energy principles to calculate the change in internal energy due to changes in internal structure because the object is actually a system

6. Describe and make qualitative and/or quantitative predictions about everyday examples of systems with internal potential energy

7. Make quantitative calculations of the internal potential energy of a system from a description or diagram of that system

8. Apply mathematical reasoning to create a description of the internal potential energy of a system from a description or diagram of the objects and interactions in that system

9. Describe and make predictions about the internal energy of systems10. Calculate changes in kinetic energy and potential energy of a system, using information from representations of

that system

Chapter 13: Oscillations About Equilibrium

Section 13-1 Periodic Motion

Repetitive or periodic motion is often described using the quantities of period or frequency.

Section 13-2 Simple Harmonic Motion

Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples include simple pendulum and mass-spring oscillator.

Mass-Spring Oscillator A spring exerts a restoring force that is proportional to the displacement from equilibrium The force of a spring is always opposite to the direction of the displacement from equilibrium A restoring force always points toward equilibrium

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Position Velocity Acceleration Net Force Description1

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Minima, maxima, and zeros of position, velocity, and acceleration are features of harmonic motion.

Sketch of Position vs. Time

Equation for Position vs. Time in SHM

Equation for Velocity vs. Time in SHM

Equation for Acceleration vs. Time in SHM

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Example 1: An air-track glider oscillates horizontally on a spring at a frequency of 0.50 Hz. Suppose the glider is pulled to the right of its equilibrium position by 12 cm and then released. Where will the glider be 1.0 s after its release? What is its velocity at this point?

Section 13-3 Connections Between Uniform Circular Motion and Simple Harmonic Motion

From our study of rotational motion, we write angular position as:

The definition of angular frequency is:

Therefore, we can write our position as a function of time as:

Similarly, we can write the equations for Velocity in Simple Harmonic Motion as:

and Acceleration in Simple Harmonic Motion as:

(Note: Students must be able to calculate force and acceleration for any given displacement for an object oscillating on a spring)

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Example 2: On December 29, 1997, a United Airlines flight from Tokyo to Honolulu was hit with severe turbulence 31 minutes after takeoff. Data from the airplane’s “black box” indicated that the 747 moved up and down with an amplitude of 30.0 m and a maximum acceleration of 1.8g. Treating the up-an-down motion of the plane as simple harmonic motion, find (a) the time required for once complete oscillation and (b) the plane’s maximum vertical speed.

Section 13-4 The Period of a Mass on a Spring

For a spring that exerts a linear restoring force, the period of a mass-spring oscillator increases with mass and decreases with spring stiffness:

Example 3: When a 0.420 kg air track cart is attached to a horizontal spring, it oscillates with a period of 0.350 s. If, instead, a different cart with mass m2 is attached to the same spring, it oscillates with a period of 0.700 s. Find (a) the force constant of the spring, and (b) the mass m2.

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Example 4: A 0.260 kg mass is attached to a vertical spring. When the mass is put into motion, its period is 1.12 s. (a) How much does the mass stretch the spring when it is at rest in its equilibrium position? (b) Suppose this experiment is repeat on a planet where the acceleration due to gravity is twice what it is on Earth. By what multiplicative factors do the period and equilibrium stretch change?

Example 5: When a mass m is attached to a vertical spring with a force constant k, it oscillates with a period T. If the spring is cut in half and the same mass is attached to it, is the period of oscillation greater than, less than, or equal to T? Justify your response.

Section 13-5 Energy Conservation in Oscillatory Motion

A system with internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy (examples include mass-spring oscillators and simple pendulums)

A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces

The work done by a conservative force is independent of then path taken. The work description is used for forces external to the system. Potential energy is used when the forces are internal interaction between parts of the system.

Changes in the internal structure can result in changes in potential energy. (examples include mass-spring oscillators and objects falling in a gravitational field)

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Equations:

The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the configuration of the objects that make up the system

Since the energy is constant in a closed system, changes in a system’s potential energy can result in changes to the system’s kinetic energy

The changes in potential and kinetic energies in a system may be further constrained by the construction of the system

For a horizontal mass spring system:

Example 6: A 500 g block is attached to a spring on a frictionless horizontal surface. The block is pulled to stretch the spring by 10 cm, then gently released. A short time later, as the block passes through the equilibrium position, its speed is 1.0 m/s. What is the block’s period of oscillation?

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Example 7: A 0.200 kg mass attached to a spring vibrates on a horizontal, frictionless table. The spring has a spring constant of 545 N/m and is initially stretched 4.50 cm and released from rest. Determine the speed of the object when the final displacement of the spring is 2.25 cm.

Example 8: A 0.980 kg block slides on a frictionless, horizontal surface with a speed of 1.32 m/s. The block encounters an unstretched spring with a force constant of 245 N/m, as shown below. (a) How far is the spring compressed before the block comes to rest? (b) For what amount of time is the block in contact with the spring before it comes to rest?

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Example 9: A bullet of mass m embeds itself in a block of mass M, which is attached to a spring of force constant k. If the initial speed of the bullet is v, find the maximum compression of the spring.

Example 10: A box of mass m is attached to a spring with a force constant k. The box rests on a horizontal, frictionless table. The spring is initially stretched to a distance A and then released from rest. The box then executes SHM characterized by a maximum speed v, an amplitude A, and an angular frequency . When the box is passing through the equilibrium position, a second box of mass m and speed v is attached to it at that instant. Discuss what happens to:

a. The maximum speed,b. The amplitude, andc. The angular frequency of the subsequent SHM.

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Section 13-6 The Pendulum

For a simple pendulum, the period increases with the length of the pendulum and decreases with the magnitude of the gravitational field

Note: This is only true for small angles.

Example 11: A grandfather clock is designed so that one swing of the pendulum in either direction takes 1.00 s. What is the length of the pendulum?

Example 12: If you look carefully at a grandfather clock, you will notice that the weight at the bootom of the pendulum can be moved up or down by turning a small screw. Suppose you have a grandfather clock at home that runs slow. Should you turn the adjusting screw so as to (a) raise the weight, or (b) lower the weight?

Example 13: A pendulum is constructed from a string 0.627 m long attached to a mass of 0.250 kg. When set in motion, the pendulum completes one oscillation every 1.59 s. If the pendulum is held at rest and the string is cut, how long will it take the mass to fall through a distance of 1.00 m?

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Example 14: Suspended from the ceiling of an elevator is a pendulum of length L. What is the period of this pendulum if the elevator accelerates upward with an acceleration a?

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