Question

In Fig. 8-25, a block slides along a track that descends through distance h. The track is frictionless except for the lower section. There the block slides to a stop in a certain distance Dbecause of friction. (a) If we decrease h, will the block now slide to a stop in a distance that is greater than, less than, or equal to D? (b) If, instead, we increase the mass of the block, will the stopping distance now be greater than, less than, or equal to D?

Short answer

1. If we decrease h, the block will slide less distance than D.
2. If we increase the mass of the block, the stopping distance will be the same.

Step-by-step solution

Given information

A figure which shows a block slides along a track that descends through distance h

To understand the concept

The problem is based on the principle of conservation of energy, which states that the total energy in an isolated system remains constant. Using the law of conservation of energy, it can be found whether the block slides to a stop at a distance that is greater than, less than, or equal to Dwhen we decrease the heightAlso, from it can be decided whether the stopping distance is greater than, less than, or equal to Dwhen we increase the mass of the block.

Formula:

Potential energy is given by,

$P.E=mgh$

Kinetic energy is given by,

$K.E=\frac{1}{2}m{v}^2$

$E=cons\tan t$

(a) To find whether the block will slide to a stop in a distance that is greater than, less than, or equal to D if we decrease h

We have,

$P.E=mgh$

For the frictionless part of the system, potential energy at the top is equal to kinetic energy at the bottom according to the law of conservation of energy. So, when we decrease the heighth, potential energy of the block will be decreased. Therefore, kinetic energy and hence velocity of the block will also be decreased. Thus, the distance traveled by the block will be less than D

Therefore, if we decrease h, the block will slide less distance than D.

(b) To find whether the stopping distance will be greater than, less than, or equal to D If, instead, we increase the mass of the block?

We have for frictionless part of the system,=

$mgh=\frac{1}{2}m{v}^2$

$v=\sqrt{2gh}$

As the velocity of the block is independent of the mass of the block, the stopping distance will be the same even if we increase the mass of the block.

Therefore, if we increase the mass of the block, the stopping distance will be the same.