This page covers the following topics:
1. Angular speed
2. Centripetal acceleration
3. Simple harmonic acceleration
4. Maxima of simple harmonic motion
For an object travelling in circular motion, the angular displacement is the angle covered by the body in radians and it is calculated using the formula θ = arc/radius. The angular displacement can then be used to find the angular speed of an object using the formula ω = θ/t. Angular speed is given in rad/s.
An object moving in circular motion is accelerating since it has a changing velocity. This is because of the fact that the direction of the object is changing since it is moving in a circle. Centripetal acceleration can be caluclated using the following formula: a = v²/r = ω²r.
Simple harmonic motion is a type of oscillation that only occurs when the following two conditions are met:
1. The acceleration of the object must be directly proportional to its displacement from its equilibrium position,
2. The acceleration must always be directed towards the equilibrium position.
The magnitude of simple harmonic acceleration can be calculated using the following formula: acceleration = (2π × frequency)² × displacement.
An object oscillating in simple harmonic motion reaches maximum speed at the equilbrium position. In this position, the acceleration of the objetct is 0. At maximum amplitude, the object has a velocity of 0, however has a maximum acceleration. The maximum speed reached by the object can be calculated using the following formula: maximum velocity = A × 2πf, where A = amplitude.
A small ball attached to a spring is moving in simple harmonic motion. Given that the frequency of its oscillation is 5 Hz and its amplitude is 7 cm, calculate the maximum acceleration of the object.
Maximum acceleration occurs at maximum amplitude.
a = (2πf)² × x
maximum acceleration = (2π × 5 Hz)² × 0.07 m = 69.1 m/s²
What is the maximum velocity reached by a ball attached to a spring oscillating at 2 Hz with amplitude of 3 cm?
v = A × 2πf
v = 0.03 m × 2π × 2 Hz
v = 0.377 m/s (to 3 significant figures)
A yo-yo is being made to move in circular motion in a circle of radius 1.2 m. Calculate the angular displacement of the yo-yo when it moves through an arclength of 80 cm.
θ = arc ÷ radius
angular displacement = 0.8 m ÷ 1.2 m = 0.667 rad (to 3 significant figures)
Calculate the frequency at which a bulb suspended from the ceiling is oscillating, given that its acceleration is 6 m/s² when it is 9 cm away from the equilibrium position.
a = (2πf)² × x
6 m/s² = (2π × f)² × 0.09 m
f = 1.30 Hz (to 3 significant figures)
Given that the velocity of a runner running on a circular track of radius 22 m is 5 m/s, calculate the angular displacement of the runner in 38 seconds to 3 significant figures.
v = rω
5 m/s = 22 m × ω
angular speed = 0.227 rad/s (to 3 significant figures)
ω = θ/t
0.227 rad/s = θ/38 s
angular displacement = 8.64 rad (to 3 significant figures)
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