Chapter 6 Problem 36

Problem

An inclined plane, fixed to the inside of an elevator, makes a 32° angle with the floor. A mass $m$ slides on the plane without friction. What is its acceleration relative to the plane if the elevator

(a) accelerates upward at 0.50 g,

(b) accelerates downward at 0.50g,

(c) falls freely, and

(d) moves upward at constant speed?

Solution

The effective value of the acceleration due to gravity in the elevator is $g_{\text{eff}}=g+a_{\text{elevator}}$. We take the upwards direction to be positive. The acceleration relative to the plane is along the plane, as shown in the free-body diagram.

(a)

$g_{\text{eff}}=g+a_{\text{elevator}}=1.5g$

$a_{\text{rel}}=g_{\text{eff}}\sin \theta =1.5g\sin 32^{\circ }=7.79{\text{m/s}}^{2}$

(b)

$g_{\text{eff}}=g+a_{\text{elevator}}=0.5g$

$a_{\text{rel}}=g_{\text{eff}}\sin \theta =0.5g\sin 32^{\circ }=2.60{\text{m/s}}^{2}$

(c)

$g_{\text{eff}}=g+a_{\text{elevator}}=0$

$a_{\text{rel}}=g_{\text{eff}}\sin \theta =0$

(d)

$g_{\text{eff}}=g+a_{\text{elevator}}=g$

$a_{\text{rel}}=g_{\text{eff}}\sin \theta =g\sin 32^{\circ }=5.19{\text{m/s}}^{2}$