Uniform circular motion is motion in a circular path with a constant speed. The time taken for one complete revolution is called the period (abbreviated as T) of the motion. Its unit of measurement is second. The number of cycles executed per second is called the frequency (abbreviated as f) of the motion.
Its unit of measurement is 1 ⁄ s which is defined to be Hertz abbreviated as Hz. Frequency and period are inverses of each other.
f = 1 ⁄ T
The number of radians executed per second is called the angular speed (abbreviated as ω) of the object.
Its unit of measurement is rad ⁄ s. Since there are 2π radians in a cycle, angular speed is equal to 2π times frequency.
ω = 2πf = 2π ⁄ T
The speed of the object (v) may be obtained as the ratio between the circumference of the circular path and the period of the motion.
v = 2πr ⁄ T = 2πrf = ωr
Example: An object is revolving in a circular path of radius 4 m with a constant speed of 10 m ⁄ s.
a) How long does it take to make one complete revolution?
Solution: r = 4 m; v = 10 m ⁄ s; T = ?
v = 2πr ⁄ T
T = 2πr ⁄ v = 2π* 4 ⁄ 10 s = 2.5 s
131 b) How many cycles does it execute per second?
Solution: f = ?
f = 1 ⁄ T = 1 ⁄ 2.5 Hz = 0.4 Hz c) Calculate its angular speed.
Solution: ω = ?
ω = 2πf = 2π* 0.4 rad ⁄ s = 2.5 rad ⁄ s
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360° thinking .
Discover the truth at www.deloitte.ca/careers
© Deloitte & Touche LLP and affiliated entities.
360° thinking .
Discover the truth at www.deloitte.ca/careers
132 7.2.1 Acceleration of a Uniform Circular Motion
Uniform circular motion is an accelerated motion even though the speed is constant because direction is changing constantly. An acceleration caused by a change in direction only is called centripetal or radial acceleration. Direction of centripetal acceleration is always towards the center of the circular path. As the particle is displaced by a small arc-length |Δs|, the position vector and the velocity vectors of the particle rotate by the same small angle. The triangle formed by the length of the initial position vector, final position vector and the arc-length |Δs|; and the triangle formed by the length of the initial velocity, final velocity vector and the vector joining the tips of this vectors, which is change in velocity Δv, are similar triangles. Thus corresponding sides of these two triangles are proportional.
|Δv| ⁄ v = |Δs| ⁄ r
|Δv| = v|Δs| ⁄ r
Dividing both sides by the time interval Δt during which this displacement took place
|Δv| ⁄ Δt = (v ⁄ r)(|Δs| ⁄ Δt)
But |Δv| ⁄ Δt is equal to the magnitude of centripetal or radial acceleration ac. And |Δs| ⁄ Δt is equal to the speed of the object v.
ac = v 2 ⁄ r
The force responsible for centripetal acceleration is called centripetal force Fc. It is related with centripetal acceleration by Newton’s second law.
Fc = mac = mv 2 ⁄ r
Example: An object of mass 5 kg is revolving in a circular path of radius 6 m with a constant speed of 4 m ⁄ s.
a) Calculate its centripetal acceleration.
Solution: m = 5 kg; r = 6 m; v = 4 m ⁄ s; ac = ?
ac = v 2 ⁄ r = 42 ⁄ 6 m ⁄ s 2 = 2.7 m ⁄ s 2
133
b) Calculate the centripetal force responsible for this acceleration.
Solution: Fc = ?
Fc = mac = 5 * 2.7 N = 13.5 N
c) Assuming the center of the circle is fixed at the origin of a coordinate system, determine the acceleration of the object when it is located at the intersection point of the circular trajectory and the positive x-axis.
Solution: West, because centripetal acceleration is always directed towards the center