Physics 1D03 - Lecture 331 Harmonic Motion ( II ) (Serway 15.2, 15.3) Mass and Spring Energy in SHM

Physics 1D03 - Lecture 332 Velocity and Acceleration a(t)  2 x(t)

Physics 1D03 - Lecture 333 Energy in SHM M Look again at the block & spring We could also write E = K+U = ½ m(v max ) 2

Physics 1D03 - Lecture 334 Question: Suppose U=½ kA 2 at t = 0, and T=1 sec. When is U=K during the cycle? (i.e. when do their curves cross?)

Physics 1D03 - Lecture 335 U, K oscillate back and forth “out of phase” with each other; the total E is constant. U K E t T

Physics 1D03 - Lecture 336 Armed with the equation of motion: and the expressions for energy: We can solve a large range of problems in SHM. Example 1: A 0.25kg block is oscillating on a spring with k=4N/m. At t=0, v=-0.2m/s and a=+0.5m/s 2. Find its total energy and the equation of motion.

Physics 1D03 - Lecture 337 Solution

Physics 1D03 - Lecture 338 Example 2: A 50.0g object connected to a spring k=35.0N/m oscillates on a horizontal, frictionless surface with an amplitude of 4.0cm. Find: a) the total energy of the system b) the speed of the object when the position is 1.0cm c) the kinetic energy when x=3.0cm d) the potential energy when x=3.0cm.

Physics 1D03 - Lecture 339 Example 3: A 200 g mass is attached to a horizontal spring and executes SHM with a period of 0.25s. The total energy of the system is 2.0J. a) Find the force constant of the spring b) Find the amplitude of the motion

Physics 1D03 - Lecture 3310 Example 4: A car having a mass of 1000kg is driven into a brick wall. The car’s bumper behaves like a spring of constant 5x10 6 N/m and compresses 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no mechanical energy loss with the wall?

Physics 1D03 - Lecture 3311 Example 5: A particle executes SHM with an amplitude of 3.0 cm. At what position does its speed equal one half of its maximum speed?

Physics 1D03 - Lecture 3312 Example 6: A uniform disc is connected to a spring and allowed to roll back and forth without slipping. What is the angular frequency of the oscillations?