# Angular acceleration and linear acceleration – problems and solutions

1. A wheel 30 cm in radius rotates at constant 5 rad/s^{2}. What is the magnitude of the linear acceleration of a point located at (a) 10 cm from the center (b) 20 cm from the center (c) on the edge of the wheel?

__Known :__

Radius (r) = 30 cm = 0.3 m

Angular acceleration (α) = 5 rad/s^{2}

__Wanted :__ linear acceleration (a) r = 0.1 m (b) r = 0.2 m (c) r = 0.3 m

__Solution :__

Relation between linear acceleration (a) and angular acceleration :

a = r α

(a) __linear acceleration, r = 0.1 m__

a = (0.1 m)(5 rad/s^{2}) = 0.5 m/s^{2}

(b)__ linear acceleration, r = 0.2 m__

a = (0.2 m)(5 rad/s^{2}) = 1 m/s^{2}

(c) __linear acceleration, r = 0.3 m__

a = (0.3 m)(5 rad/s^{2}) = 1.5 m/s^{2}

2. A pulley 50 cm in radius. If the linear acceleration of a point located on the edge of the pulley is 2 m/s^{2}, determine the angular acceleration of the pulley!

__Known :__

Radius (r) = 50 cm = 0,5 m

linear acceleration (a) = 2 m/s^{2}

__Wanted :__ the angular acceleration

__Solution :__

α = a / r = 2 / 0.5 = 4 rad/s^{2 }

3. The blades in a blender 20 cm in radius, initially at rest. After 2 seconds, the blades rotates 10 rad/s. Determine the magnitude of linear acceleration (a) a point located at 10 cm from the center (b) a point located at the edge of the blades.

__Known :__

Radius (r) = 20 cm = 0.2 m

The initial angular velocity (ω_{o}) = 0

The final angular velocity (ω_{t) }= 10 radians/second

Time interval (t) = 2 seconds

__Wanted :__ the linear acceleration of a point located at (a) r = 0.1 m (b) r = 0.2 m

__Solution :__

ω_{t} = ω_{o }+ α t

10 = 0 + α (2)

10 = 2 α

α = 10 / 2

α = 5 rad/s

(a) __linear acceleration of r = 0.1 m__

a = r α = (0.1 m)(5 rad/s^{2}) = 0.5 m/s^{2}

(b)__ linear acceleration of r = 0.2 m__

a = r α = (0.2 m)(5 rad/s^{2}) = 1 m/s^{2}

4. A wheel 20 cm in radius is accelerated for 2 seconds from 20 rad/s to rest. Determine the magnitude of linear acceleration (a) a point located at 10 cm from the center (b) a point located at 10 cm from the center.

__Known :__

Radius (r) = 20 cm = 0.2 m

The initial angular speed (ω_{o}) = 20 rad/s

The final angular speed (ω_{t}) = 0

Time interval (t) = 2 seconds

__Wanted :__ The linear acceleration (a) r = 0.1 m (b) r = 0.2 m

__Solution :__

ω_{t }= ω_{o }+ α t

0 = 20 + α (2)

-20 = 2 α

α = -20 / 2

α = -10 rad/s

Negative sign mean the angular speed is decrease.

(a) __linear acceleration of r = 0.1 m__

a = r α = (0.1 m)(-10 rad/s^{2}) = -1 m/s^{2}

(b) __linear acceleration of r = 0.2 m__

a = r α = (0.2 m)(-10 rad/s^{2}) = -2 m/s^{2}

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