Centripetal force – problems and solutions

1. A 200-gram ball, attached to the end of a cord, is revolved in a horizontal circle with an angular speed of 5 rad s^-1. If cord’s length is 60 cm, what is the centripetal force?

Known :

Object’s mass (m) = 200 gr = 200/1000 kg = 2/10 kg = 0.2 kg

Angular speed (ω) = 5 rad/s

Cord’s length = radius (r) = 60 cm = 60/100 m = 0.6 m

Wanted : The centripetal force

Solution :

The centripetal force is the resultant force that causes the centripetal acceleration.

The equation of the centripetal force :

∑F = m a_s

∑F = m v²/r = m ω² r

∑F = Centripetal force, m = object’s mass, v = linear velocity, ω = angular velocity, r = radius.

∑F = m ω² r = (0.2)(5)² (0.6) = (0.2)(25)(0.6) = 3 N

2. A stone attached at the end of a cord and rotated in a horizontal circle by a student. If the final speed of the stone = 2 x the initial speed, then what is the centripetal force.

Known :

Stone’s mass = m

Stone’s speed = v

Cord’s length = radius = r

Wanted: The centripetal force

Solution :

3. A curve road of radius R is designed so that a car traveling at speed 10 ms^–1 can negotiate the turn safely. The coefficient of static friction between car and road = 0.5. What is the radius? Acceleration due to gravity (g) = 10 ms^–2.

Known :

Speed (v) = 10 m/s

The coefficient of static friction between car and road (μ_s) = 0.5

Acceleration due to gravity (g) = 10 m/s²

Wanted: Radius

Solution :

The only one force in the horizontal direction is the force of static friction. The equation of the static friction :

4. The coefficient of static friction between tire and road is 0.4. If acceleration due to gravity is 10 m/s², what is the maximum speed so the car can turn without skidding out of a curved path.

Known :

Coefficient of static friction (μ_s) = 0.4

Acceleration due to gravity (g) = 10 m/s²

Radius of path (R) = 40 meters

Wanted: maximum speed (v)

Solution :

The equation of Newton’s second law in uniform circular motion :

ΣF = centripetal force = net force, m = mass, a_s = centripetal acceleration, v = linear speed, R = radius of path

Centripetal force

Centripetal force is the net force which produces centripetal accelerations. In this case, the centripetal force is the force of static friction.

The equation of the force of static friction :

μ_s = coefficient of static friction, w = weight, m = mass, g = acceleration due to gravity

The maximum speed (v) :