The two blocks (with m=16 kg and M= 88 kg ) shown in Fig. 6-56 are not attached. The coefficie

step 1 All right. So in this problem, we've got these two masses that are being pushed against each oth

The two blocks (with m=16 kg and M= 88 kg ) shown in Fig....

The two blocks (with $m=16 \mathrm{~kg}$ and $M=$ $88 \mathrm{~kg}$ ) shown in Fig. 6-56 are not attached. The coefficient of static friction between the blocks is $\mu^{\text {stat }}=$ $0.38$, but the surface beneath the larger block is frictionless. What is the minimum magnitude of the horizontal force $\vec{F}$ required to keep the smaller block from slipping down the larger block?

Key Concepts

Horizontal Acceleration Effect

The horizontal force applied to the larger block causes it to accelerate, which in turn influences the smaller block through the pseudo force in the non-inertial frame. This acceleration must be sufficient such that the inertial effects combine with gravity to create an effective gravitational field that the static friction can counteract, thereby preventing the smaller block from slipping down.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two contacting surfaces. In this scenario, the static friction between the two blocks must be strong enough to counteract the component of gravitational force pulling the smaller block down the incline of the larger block. The maximum static frictional force is determined by the product of the coefficient of static friction and the normal force between the blocks.

Free-Body Diagram Analysis

Drawing a free-body diagram for the small block is essential in identifying all the forces acting on it. This diagram typically includes gravity, the pseudo force due to acceleration, and the frictional force. By resolving these forces into components along and perpendicular to the surface of the larger block, one can determine the conditions necessary for equilibrium and prevent the block from sliding.

Pseudo Force

In a non-inertial frame, pseudo forces are introduced to account for the acceleration of the reference frame. This force acts on objects in the opposite direction to the acceleration of the frame. In this problem, the pseudo force, when combined with gravity, creates an effective force vector that determines the conditions under which the smaller block remains stationary relative to the larger block.

Non-Inertial Reference Frame

Analyzing the problem from the perspective of the accelerating larger block transforms the situation into a non-inertial frame. In this frame, an additional fictitious force (or pseudo force) appears, which needs to be accounted for alongside gravity. This approach simplifies the problem by allowing one to consider the relative motion of the smaller block under the combined effects of gravity and the inertial force resulting from the acceleration.

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