A uniform cube of side length 8.0 cm rests on a horizontal floor. The coefficient of static fric

step 1 In this problem on the topic of equilibrium and elasticity, we are told that a uniform cube, whi

A uniform cube of side length 8.0 cm rests on a horizontal...

A uniform cube of side length 8.0 $\mathrm{cm}$ rests on a horizontal floor. The coefficient of static friction between cube and floor is $\mu .$ A horizontal pull $\vec{P}$ is applied perpendicular to one of the vertical faces of the cube, at a distance 7.0 $\mathrm{cm}$ above the floor on the vertical midline of the cube face. The magnitude of $\vec{P}$ is gradually increased. During that increase, for what values of $\mu$ will the cube eventually (a) begin to slide and (b) begin to tip? (Hint: At the onset of tipping, where is the normal force located?)

Key Concepts

Static Friction

Static friction is the force that resists the initiation of sliding motion between two contacting surfaces. It is proportional to the normal force and is characterized by the coefficient of static friction, ?. In problems involving potential motion of a rigid body, it sets the threshold force required to overcome resistance to sliding when an external horizontal force is applied.

Torque and Rotational Equilibrium

Torque, or the moment of force, is a measure of the turning effect produced by a force acting at a distance from a pivot point. Rotational equilibrium occurs when the sum of all torques about any axis is zero. In many rigid body problems, including those involving tipping, analyzing the torques due to applied forces and the weight of the object (acting through the center of mass) is essential to determine when rotation or tipping will occur.

Tipping Conditions

Tipping occurs when the moments produced by applied forces overcome the stabilizing moment provided by gravity acting through the center of mass. At the onset of tipping, the normal force shifts to the edge of the base of support, and the object begins to rotate about that edge. Determining the moment balance about this pivot point allows one to assess the conditions under which tipping, rather than sliding, is the primary mode of motion.

Center of Mass

The center of mass is the point at which the mass of an object can be considered to be concentrated for the analysis of translational and rotational motion. For uniform solids, such as a cube, the center of mass is located at its geometric center. The position of the center of mass is critical in calculating the gravitational torque about a pivot point and understanding the stability of the object under applied forces.

Force Application Point

The point at which an external force is applied affects the moment arm and, consequently, the torque produced. In problems where forces are applied away from the center of mass, the height or lateral position of the force application can significantly influence whether the object will begin to slide or tip. A higher application point increases the moment arm for tipping, making rotation more likely at lower values of the applied force.

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