Derivation of linear speed and angular velocity By definition: v = s
__ t
From θ = s v = rθ Since ω = θ
__ t :
v = rω QUESTIONS AND ANSWERS
4. A record turntable is rotating at 78 revolutions per minute (rpm). Calculate its angular velocity in rad s–1.
Solution 78 rpm = 78
____ 60
= 1.3 revolutions per second = (1.3)(2π) = 2.6π rad s–1
5. A train turntable has a radius of 20 m. It completes two full turns in 5 minutes. A driver is standing on the circumference of the turntable as it turns. Calculate the angular velocity in rad s–1 and speed of the driver in m s–1.
Solution
2 full rotations = 4π rad 5 minutes = 300 seconds
Using ω = θ
__ t gives ω = 4π
_____ 300 = 0.0419 rad s–1
From v = rω, v = (20)(0.0419) = 0.838 m s–1
6. An object moving in a circular motion of radius 10 m, at a uniform velocity, is capable of travelling 2 km in 5 minutes. Calculate (i) its linear speed and (ii) its angular velocity.
Solution (i) Speed = 2 km in 5 minutes = 2000
= 6 2 (ii) Radius = 10 m: ω = v You may now complete Exercise 8B (page 80). Centripetal force
A centripetal force (F) is the force directed towards the centre of a circle that is necessary to keep a body moving in a circular path. Centripetal forces, like other forces, are measured in newtons and can be calculated by multiplying mass and acceleration. However, we need to specify a centripetal acceleration for this force rather than the usual linear acceleration.
If you apply a force perpendicular to a body’s direction of movement, it will turn in a circle. That is why the centripetal force is always applied towards the centre of the circle. From Fig. 8.4, you can see the combination of tangential speed and centripetal force cause the particle to move in a circle. It should be noted that centripetal force is just a force that causes an object to move in a circular motion.
The force directed towards the centre of a circle that is necessary to keep a body moving in a circular path is called the centripetal force (F).
