A student connects an object with mass m to a rope with a length r and then rotates the rope around her head parallel to the ground. The object takes 0.5 seconds to complete one rotation. Mass = 50 gLength of rope = 1.2 m
a) What is the object’s speed of rotation?
b) What is the object’s centripetal acceleration?
c) What tension force is required to maintain this motion?

(a) ω = 12.57 rad/s

(b) a = 189.5 m/s²

(c) T = 9.47 N

Explanation:

(a)

The speed of rotation is given by the formula:

ω = θ/t

where,

ω = speed of rotation = ?

θ = angular displacement = (1 rotation)(2π rad/1 rotation) = 2π rad

t = time taken = 0.5 s

Therefore,

ω = 2π rad/0.5 s

ω = 12.57 rad/s

(b)

The centripetal acceleration of the object is given by the formula:

a = v²/r

where,

a = Centripetal Acceleration = ?

v = linear speed of object = rω

r = length of rope = 1.2 m

Therefore,

a = (rω)²/r

a = rω²

a = (1.2 m)(12.57 rad/s)²

a = 189.5 m/s²

(c)

The tension required to maintain the motion is equal to the centripetal force:

Tension = Centripetal Force

T = ma

where,

m = mass of object = 50 g = 0.05 kg

Therefore,

T = (0.05 kg)(189.5 m/s²)

T = 9.47 N