A block of mass rests on an adjustable rough incline

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

Problem

A block of mass

m

rests on an adjustable rough incline. The coefficient of static friction between the block and the surface is

\mu_s

, and the coefficient of kinetic friction is

\mu_k

(

\mu_k < \mu_s

). The angle

\theta

of the incline can be changed slowly. The block is released from rest at each angle. (a) Derive an expression for the minimum angle

\theta_{\text{min}}

at which the block just begins to slide. (b) Evaluate

\theta_{\text{min}}

numerically for

\mu_s = 0.40

. (c) Once the block is sliding, the incline is quickly fixed at

\theta = 40^{\circ}

. Determine the magnitude of the block’s acceleration down the incline, expressing your answer in terms of

g

\theta

, and

\mu_k

and then computing a numerical value for

\mu_k = 0.30

. (d) Briefly discuss how increasing

\mu_s

would affect

\theta_{\text{min}}

and the physical reasoning behind this effect.