Simple Harmonic Motion. An Introduction. Periodic Motion. Repeated motion along the same path over a fixed reproducible period of time Acrobat on a trapeze Wrecking ball Playground swing Mass attached to a spring. x. m. L. x = -A. x = 0. x = +A. Oscillators.
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Repeated motion along the same path over a fixed reproducible period of time
Law of Conservation of Energy
FExample 1:The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion?
Period: T = 0.500 s
Frequency: f = 2.00 Hz
A restoring force directly proportional to displacement is responsible for the motion of a spring.
F = -kx
F: restoring force
k: force constant
x: displacement from equilibrium
k = =A 4-kg mass suspended from a spring produces a displacement of 20 cm. What is the spring constant?
The stretching force is the weight (W = mg) of the 4-kg mass:
F = (4 kg)(9.8 m/s2) = 39.2 N
Now, from Hooke’s law, the force constant k of the spring is:
k = 196 N/m
mThe mass m is now stretched a distance of 8 cm and held. What is the potential energy? (k = 196 N/m)
The potential energy is equal to the work done in stretching the spring:
U = 0.627 J
A 2-kg mass hangs at the end of a spring whose constant is k = 400 N/m. The mass is displaced a distance of 12 cm and released. What is the acceleration at the instant the displacement is x = +7 cm?
a = -14.0 m/s2
Note: When the displacement is +7 cm (downward), the acceleration is -14.0 m/s2 (upward) independent of motion direction.
mWhat is the maximumacceleration for the 2-kg mass in the previous problem? (A = 12 cm, k = 400 N/m)
The maximum acceleration occurs when the restoring force is a maximum; i.e., when the stretch or compression of the spring is largest.
F = ma = -kx
amax = ± 24.0 m/s2
x = 0
x = -0.2 m
x = +0.2 m
The frictionless system shown below has a 2-kg mass attached to a spring (k = 400 N/m). The mass is displaced a distance of 20 cm to the right and released.What is the frequency of the motion?
f = 2.25 Hz
L = 0.993 mWhat must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)?