Physics 7E Prof. D. Casper

Course Stuff MasteringPhysics: Tutoring Center First long homework assignment (Chapter 12) is due Thursday (by 7 am) Tutoring Center Open M – Th, 6 – 9:30 pm in Rowland Hall 158 and 160 starting tonight! Drop-in free tutoring

Simple Harmonic Motion The equation of motion includes 3 independent constants that combine to describe the motion: 𝑥 𝑡 =𝐴 cos⁡(𝜔𝑡+𝜙) The angular frequency 𝜔 Determined by the properties (force equation) of the system itself The amplitude 𝐴 and the phase constant 𝜙 Determined by the starting conditions of the motion Demo

Parameter Summary Angular frequency/period/frequency Amplitude Determined by the force equation of system Expresses the time required for one cycle or repetition Amplitude Determined by starting conditions Maximum distance from equilibrium Phase constant Where is the system in the cycle at time t = 0?

Phase The cosine function in 𝑥 𝑡 =𝐴 cos⁡(𝜔𝑡+𝜙) acts on an angle In radians (important!) The phase 𝜔𝑡+𝜙 expresses the system’s progress through its repeating cycle The phase is 𝜙 at 𝑡=0 The phase increases linearly (slope = 𝜔) over time A change of 2𝜋 radians in the phase means… One complete cycle No difference in the physical state Systems whose phases differ by a multiple of 2𝜋 are indistinguishable “In phase”

Working with Simple Harmonic Motion Position: 𝑥 𝑡 =𝐴 cos⁡(𝜔𝑡+𝜙) Maximum displacement: ±𝐴 What is the velocity? 𝑣 𝑡 = 𝑥 (𝑡)=−𝐴𝜔 sin⁡(𝜔𝑡+𝜙) Maximum velocity: ±𝐴𝜔 What is the acceleration? 𝑎 𝑡 = 𝑣 (𝑡)= 𝑥 (𝑡)=−𝐴 𝜔 2 cos 𝜔𝑡+𝜙 =− 𝜔 2 𝑥(𝑡) Maximum acceleration: ±𝐴 𝜔 2 Remember: SHO condition is 𝑥 =− 𝜔 2 𝑥

B. T remains the same and vmax doubles. Q14.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object’s maximum speed vmax? A. T and vmax both double. B. T remains the same and vmax doubles. C. T and vmax both remain the same. D. T doubles and vmax remains the same. E. T remains the same and vmax increases by a factor of 2 Answer: B

A14.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object’s maximum speed vmax? A. T and vmax both double. B. T remains the same and vmax doubles. C. T and vmax both remain the same. D. T doubles and vmax remains the same. E. T remains the same and vmax increases by a factor of 2 .

This is an x-t graph for an object in simple harmonic motion. Q14.2 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negative velocity vx? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T Answer: A

A14.2 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negative velocity vx? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T

This is an x-t graph for an object in simple harmonic motion. Q14.3 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negative acceleration ax? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T Answer: D

A14.3 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negative acceleration ax? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T (or t = 0)

Energy in Simple Harmonic Motion Since we can describe the position and velocity of an object in Simple Harmonic Motion, we also know its kinetic energy and its potential energy For a mass on a spring: 𝐾 𝑡 = 1 2 𝑚 𝑣 2 = 1 2 𝑚 𝜔 2 𝐴 2 sin 2 ⁡(𝜔𝑡+𝜙) 𝑈 𝑡 = 1 2 𝑘 𝑥 2 = 1 2 𝑘 𝐴 2 cos 2 (𝜔𝑡+𝜙) but remember 𝜔 2 = 𝑘 𝑚 , so 𝑈 𝑡 = 1 2 𝑚 𝜔 2 𝐴 2 cos 2 (𝜔𝑡+𝜙) 𝐸 𝑡 =𝐾 𝑡 +𝑈 𝑡 = 1 2 𝑚 𝜔 2 𝐴 2 cos 2 𝜔𝑡+𝜙 + sin 2 𝜔𝑡+𝜙 E(t)= 1 2 m 𝜔 2 𝐴 2 = 1 2 𝑘 𝐴 2 The total mechanical energy is constant (and conserved)

Energy Conservation

E. more than one of the above Q14.6 This is an x-t graph for an object connected to a spring and moving in simple harmonic motion. At which of the following times is the potential energy of the spring the greatest? t = T/8 t = T/4 C. t = 3T/8 D. t = T/2 E. more than one of the above Answer: D

E. more than one of the above This is an x-t graph for an object connected to a spring and moving in simple harmonic motion. At which of the following times is the potential energy of the spring the greatest? t = T/8 t = T/4 C. t = 3T/8 D. t = T/2 E. more than one of the above

E. more than one of the above Q14.7 This is an x-t graph for an object connected to a spring and moving in simple harmonic motion. At which of the following times is the kinetic energy of the object the greatest? t = T/8 t = T/4 C. t = 3T/8 D. t = T/2 E. more than one of the above Answer: B

E. more than one of the above This is an x-t graph for an object connected to a spring and moving in simple harmonic motion. At which of the following times is the kinetic energy of the object the greatest? t = T/8 t = T/4 C. t = 3T/8 D. t = T/2 E. more than one of the above

The Simple Pendulum

Simple Pendulum Period 𝐼 𝜃 =𝑚 𝐿 2 𝜃 =−𝑚𝑔 sin 𝜃 ×𝐿 𝜃 =− 𝑔 𝐿 sin 𝜃 ≈− 𝑔 𝐿 𝜃 𝜔 2 = 𝑔 𝐿 Angular frequency, period, etc depend only on length of pendulum (and g) No dependence on the mass

Physical Pendulum Identical restoring force Any shape and moment of inertia I 𝐼 𝜃 =−𝑚𝑔 sin 𝜃 ×𝑑≈−𝑚𝑔𝑑 𝜃 𝜃 =− 𝑚𝑔𝑑 𝐼 𝜃 𝜔 2 = 𝑚𝑔𝑑 𝐼