Centripetal acceleration – problems and solutions

1. A ball, attached to the end of a horizontal cord, is revolved in a circle of radius 20 cm. The ball around 360^o each second. Determine the magnitude of the centripetal acceleration!

Known :

Angular speed (ω) = 360^o/second = 1 revolution/second = 6.28 radians/second

Radius (r) = 20 cm = 0.2 m

Wanted : Centripetal acceleration (a_r)

Solution :

a_r = v² / r —> v = r ω

a_r = (r ω)² / r = r² ω² / r

a_r = r ω²

a_s = centripetal acceleration, v = linear velocity, r = radius, ω = angular velocity

The magnitude of the centripetal acceleration :

a_r = r ω² a_r = (0,2 m)(6.28 rad/s)

a_r = 1.256 m/s²

2. A wheel 30 cm in radius rotate at a rate of 180 rpm. Determine the centripetal acceleration of a point on the edge of wheel!

Known :

Radius (r) = 30 cm = 0.3 m

Angular speed (ω) = 180 revolutions / 60 seconds = 3 revolutions / second = (3)(6.28 radians) / second = 18.84 radians/second

Wanted : centripetal acceleration (a_r) of r = 0.3 m

Solution :

The magnitude of the centripetal acceleration :

a_r = r ω²

a_r = (0.3 m)(18.84 rad/s)

a_r = 5.65 m/s²

3. A race car moving on a circular track of radius 50 meters. If car’s speed is 72 km/h, determine the magnitude of the centripetal acceleration!

Known :

Radius (r) = 50 meters

Speed (v) = 72 km/h = (72)(1000 meters) / 3600 seconds = 20 meters/second

Wanted : the magnitude of the centripetal acceleration (a_r)

Solution :

a_r = v² / r = 20² / 50 = 400 / 50 = 8 m/s²

4. A car has the maximum centripetal acceleration 10 m/s², so the car can turn without skidding out of a curved path. If the car is moving at a constant 108 km/h, what is the radius of unbanked curve ?

Known :

Centripetal acceleration (a_r) = 10 m/s²

Car’s speed (v) = 108 km/h = (108)(1000) / 3600 = 30 meters/second

Wanted : radius (r)

Solution :

r = v² / a_r

r = 30² / 10 = 900 / 10 = 90 meters

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