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Chapter 13: Problem 46

Blocks \(A\) and \(B\) each have a mass \(m\). Determine the largest horizontal force \(\mathbf{P}\) which can be applied to \(B\) so that \(A\) will not move relative to \(B\). All surfaces are smooth.

Short Answer

Expert verified
The largest horizontal force \(\mathbf{P}\) which can be applied to block \(B\) so that \(A\) will not move relative to \(B\) is 0 Newton.

Step by step solution

01

Understanding the Situation

First, understand the situation. We have two blocks, A and B, on top of each other. A force \(P\) is applied to block B. For block A not to slide off block B, the frictional force between the two blocks must be enough to balance out the horizontal force \(P\).
02

Applying Newton's Second Law

Apply Newton's Second Law on block A. For block A to stay in equilibrium, the net force on it should be zero. Hence, the frictional force \(f\) equals the applied force \(P\). (i.e., \(f = P\))
03

Calculate the frictional force

Because all surfaces are smooth, the frictional force between the blocks is only due to the normal force between them. The normal force is equal to the weight of block A which equals \(mg\). The frictional force \(f\) between two bodies when one is not moving relative to the other (also known as static friction) can be calculated using the equation \(f = \mu n\) where \(\mu\) is the coefficient of static friction and \(n\) is the normal force. In this problem, as there are smooth surfaces, \(\mu\) is zero. Thus, the frictional force \(f\) between blocks A and B is equal to \(0*mg=0\).
04

Determine the largest Horizontal Force

Since the static frictional force (f) and the force applied (P) are equal and the frictional force (f) is zero, the force \(P\) should be zero for block A to not move relative to block B.

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Most popular questions from this chapter

The 0.5-lb ball is guided along the vertical circular path \(r=2 r_{c} \cos \theta\) using the \(\operatorname{arm} O A\). If the arm has an angular velocity \(\dot{\theta}=0.4 \mathrm{rad} / \mathrm{s}\) and an angular acceleration \(\ddot{\theta}=0.8 \mathrm{rad} / \mathrm{s}^{2}\) at the instant \(\theta=30^{\circ}\) determine the force of the arm on the ball. Neglect friction and the size of the ball. Set \(r_{c}=0.4 \mathrm{ft}\).

The collar has a mass of \(2 \mathrm{~kg}\) and travels along the smooth horizontal rod defined by the equiangular spiral \(r=\left(e^{\theta}\right) \mathrm{m}\), where \(\theta\) is in radians. Determine the tangential force \(F\) and the normal force \(N\) acting on the collar when \(\theta=45^{\circ}\), if the force \(F\) maintains a constant angular motion \(\dot{\theta}=2 \mathrm{rad} / \mathrm{s}\) Prob. 13-104

Block \(A\) has a weight of \(8 \mathrm{lb}\) and block \(B\) has a weight of \(6 \mathrm{lb}\). They rest on a surface for which the coefficient of kinetic friction is \(\mu_{k}=0.2\). If the spring has a stiffness of \(k=20 \mathrm{lb} / \mathrm{ft}\), and it is compressed \(0.2 \mathrm{ft}\), determine the acceleration of each block just after they are released. Prob. 13-13

The collar, which has a weight of 3 lb, slides along the smooth rod lying in the horizontal plane and having the shape of a parabola \(r=4 /(1-\cos \theta)\), where \(\theta\) is in radians and \(r\) is in feet. If the collar's angular rate is constant and equals \(\dot{\theta}=4 \mathrm{rad} / \mathrm{s}\), determine the tangential retarding force \(P\) needed to cause the motion and the normal force that the collar exerts on the rod at the instant \(\theta=90^{\circ}\). Prob. 13-108

The conveyor belt is moving downward at \(4 \mathrm{~m} / \mathrm{s}\). If the coefficient of static friction between the conveyor and the \(15-\mathrm{kg}\) package \(B\) is \(\mu_{s}=0.8\), determine the shortest time the belt can stop so that the package does not slide on the belt. Prob. \(13-27\)
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