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Chapter 8: Q. 54 (page 201)

FIGURE P8.54 shows a small block of mass m sliding around the inside of an L-shaped track of radius r. The bottom of the track is frictionless; the coefficient of kinetic friction between the block and the wall of the track is 碌k,The block's speed is vo at to =0 Find an expression for the block's speed at a later time t.

Short Answer

Expert verified

The expression for the velocity at time t is vt=(vor)(vo渭kt+r)

Step by step solution

01

Given Information

mass= m

radius= r

The bottom of the track is frictionless

coefficient of kinetic friction between the block and the wall of the track is 碌k,

speed is vo at to =0

02

Explanation

Centripetal force is given by

Fc=mv2r
where m = mass, v = velocity and r = radius.
Frictional force is 碌kN

Where 碌k,= coefficient of friction and N is Normal force

From the Newton's second law, we know

mdvdt=-渭k(mv2r)dvdt=-渭kv2r

Now integrate and find vt

鈭�v0vt-dvv2=鈭�0t渭kdtr1vt-1v0=渭ktr1vt=渭ktr+1v0vt=vorvo渭kt+r

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