Chapter 8, Problem 103A

(a)

To determine

To Explain: The condition under which the centripetal acceleration would be zero.

Explanation of Solution

Introduction:

The centripetal acceleration occurs when an object is moving in a circular path.The centripetal acceleration always tends to move the object towards the center of the circle. Its magnitude is defined as the ratio of square of the speed (of a body) along the curve to the distance from the center of the circle.

The centripetal acceleration is always oriented inwards along the radial direction of circulatory motion.

Mathematically, the centripetal acceleration

  $a_c=\frac{v^2}{r}$

For uniform circular motion of the object, the circular radius is constant, and the centripetal acceleration is zero.

Conclusion: 

The centripetal acceleration is zero under the conditions of uniform circular motion.

(b)

To determine

To Explain:The conditions that the linear tangential acceleration is zero.

Explanation of Solution

Introduction:

Tangential acceleration is defined as is the measure of change in the tangential velocity of a point for a certain radiuswith the change in time.

The normal acceleration causes the motion to be circular. The tangential acceleration can make the rotation slower or faster.

The uniform circular motion can be definedas the movement of an object with a constant velocity around a fixed point which means the rotation is at a constant speed. In such case, the tangential component of acceleration will be zero.The velocity vector changes the direction but not magnitude. So, that tangential acceleration is zero.

Conclusion: 

When the rotational speed changes, the tangential acceleration is zero for uniform circular motion.

(c)

To determine

To Explain:Whether the tangential acceleration would be non-zero while the centripetal acceleration is zero.

Explanation of Solution

Introduction:

The tangential acceleration is caused by change in the magnitude of velocity and the centripetal acceleration is caused by the change in direction of motion.

A uniform circular motion of a body is characterized by theconstant tangential velocity. Its direction is always inwards along the radius and pointing towards the centripetal acceleration.

The non-uniform circular motion defines as the change in the speed while moving in circular path. This change causes change in the tangential acceleration. Centripetal acceleration exists only when the object moves in the circular path. Thus, for the non-uniform circular path, centripetal acceleration becomes zero.

Conclusion:

In non-uniform circular motion, thetangential acceleration could be non-zero while the centripetal acceleration is zero.

(d)

To determine

To Explain: Whether the centripetal acceleration could be non-zero while the tangential acceleration be zero.

Explanation of Solution

Introduction:

The tangential acceleration is caused by change in the magnitude of velocity and the centripetal acceleration is caused by the change in direction of motion.

The direction of velocity always changes while moving in a circular path. The direction of change of the velocity of the object is always towards the center of circle when rolling around it which means a non-zero component of the acceleration which is pointing towards radially inwards, When an objects moves in a circular motion, the non-zero centripetal acceleration exists. For having zero tangential acceleration on the object, it should rotate on the path with the constant speed.

Thus, when an object is moving with constant speed in a circle which is not varying throughout, only then there is an existence of centripetal acceleration.

Conclusion: 

The non-zero centripetal acceleration exists for an object when the object is moving with a fixed speed.