A block of mass is to be pushed along a horizontal rough floor by a force that makes…

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

Problem

A block of mass

m

is to be pushed along a horizontal rough floor by a force

P

that makes an angle

\theta

below the horizontal with the block on the verge of motion. (i) Establish an expression for the minimum

P

required in terms of

m

g

\mu_s

and

\theta

. (ii) Investigate how

P

varies with

\theta

for

0^{\circ} < \theta < 90^{\circ}

. Show that

P

increases monotonically with

\theta

in this range. (iii) Hence prove that the least value of

P

occurs when

\theta = 0^{\circ}

, i.e. when the push is exactly horizontal. (iv) Taking

m = 15\,\text{kg}

and

\mu_s = 0.25

, compute this least value of

P

and compare it with the force needed when the same push is applied at

20^{\circ}

below the horizontal.