# 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}** = v**^{2}** / r** —> v = r ω

a_{r} = (r ω)^{2 }/ r = r^{2 }ω^{2} / r

**a**_{r}** = r ****ω**^{2}** **

*a*_{s}* = centripetal acceleration, v = linear velocity, r = radius, **ω **= angular velocity*

__The magnitude of the centripetal acceleration __:

a_{r} = r ω^{2 }a_{r} = (0,2 m)(6.28 rad/s)

a_{r} = 1.256 m/s^{2}

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 ω^{2 }

a_{r }= (0.3 m)(18.84 rad/s)

a_{r} = 5.65 m/s^{2}

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} = v^{2} / r = 20^{2} / 50 = 400 / 50 = 8 m/s^{2}

4. A car has the maximum centripetal acceleration 10 m/s^{2}, 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/s^{2}

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

__Wanted :__ radius (r)

__Solution :__

r = v^{2} / a_{r}

r = 30^{2} / 10 = 900 / 10 = 90 meters

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