On a certain laboratory surface the coefficient of kinetic friction between a block…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Force and Translational Dynamics · Force and Translational Dynamics

Problem

On a certain laboratory surface the coefficient of kinetic friction between a block and the surface depends on the block’s speed according to

\mu_{k}(v) = \mu_{0} + k v

, where

\mu_{0}

and

k

are positive constants and

v

is the speed of the block. A student pulls a

10\,\text{kg}

block along this surface with a constant horizontal force of magnitude

30\,\text{N}

. The constants are

\mu_{0} = 0.12

and

k = 0.004\,\text{s}\!/\!\text{m}

. (a) Derive an expression for the block’s steady-state speed

v_{s}

(the speed at which it moves at constant velocity) in terms of

F_{p}

m

g

\mu_{0}

, and

k

. (b) Calculate the numerical value of

v_{s}

for the given parameters, taking

g = 9.8\,\text{m/s}^{2}

. (c) Discuss the behavior of your expression for

v_{s}

as the pulling force

F_{p}

approaches the value

\mu_{0} m g

from above. Explain physically why this limiting behavior is expected.