A bookcase weighing 1500 N rests on a horizon- tal surface for which the coefficient of static f

step 1 By balancing the torque about the left end equals 0, we find that FL times 5, we will be balanci

A bookcase weighing 1500 N rests on a horizon- tal surface...

A bookcase weighing 1500 $\mathrm{N}$ rests on a horizon- tal surface for which the coefficient of static friction is $\mu_{\mathrm{s}}=0.40 .$ The bookcase is 1.80 $\mathrm{m}$ tall and 2.00 $\mathrm{m}$ wide; its center of gravity is at its geo- metrical center. The bookcase rests on four short legs that are each 0.10 $\mathrm{m}$ from the edge of the bookcase. A person pulls on a rope attached to an upper corner of the bookcase with a force $\vec{\boldsymbol{F}}$ that makes an angle $\theta$ with the bookcase (Fig. $P 11.97 )$ . (a) If $\theta=90^{\circ},$ so $\vec{F}$ is horizontal, show that as $F$ is increased from zero, the bookcase will start to slide before it tips, and calculate the magnitude of $\vec{\boldsymbol{F}}$ that will start the bookcase sliding. (b) If $\theta=0^{\circ},$ so $\vec{\boldsymbol{F}}$ is vertical, show that the bookcase will tip over rather than slide, and calculate the magnitude of $\vec{\boldsymbol{F}}$ that will cause the bookcase to start to tip. (c) Calculate as a function of $\theta$ the magnitude of $\vec{\boldsymbol{F}}$ that will cause the bookcase to start to slide and the magnitude that will cause it to start to tip. What is the smallest value that $\theta$ can have so that the bookcase will still start to slide before it starts to tip?

Key Concepts

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. Its maximum value is given by the product of the static friction coefficient and the normal force. In problems involving objects at rest, such as the bookcase on a floor, understanding static friction is essential to determine the force needed to overcome this resistance and start sliding, especially when external forces are applied.

Torque and Rotational Equilibrium

Torque, or moment of force, is the measure of the force's ability to cause an object to rotate about a pivot point. In the analysis of tipping, it is crucial to evaluate the torques produced by various forces acting at different distances from the pivot. Equilibrium is maintained until the sum of the torques about the tipping point exceeds a critical value, leading to rotation or tipping over. This concept is central to determining whether an applied force will cause an object to tip rather than simply slide.

Center of Mass

The center of mass of an object is the point at which its mass is considered to be concentrated. For uniformly distributed objects, this is typically at the geometric center. The location of the center of mass affects both the stability against tipping and the way forces such as gravity contribute to torques. In stability problems, the relative position of the center of mass with respect to the pivot point helps determine the conditions under which the object will tip.

Force Decomposition

Force decomposition involves breaking a force into components, typically horizontal and vertical. When a force is applied at an angle, its components can have different effects on the motion of an object. The horizontal component may contribute to sliding by overcoming friction, while the vertical component can alter the normal force and thus the frictional resistance, or induce additional torque that could lead to tipping. Understanding this decomposition is key to analyzing the effects of forces applied at oblique angles.

Tipping vs. Sliding Transition

The transition between tipping and sliding is determined by the interplay between the applied force, frictional forces, and the torques about a pivot point. An object will start to slide when the horizontal force component exceeds the maximum static friction, but it will tip when the applied force creates a torque that overcomes the restoring moment due to gravity about the edge of the object’s base. Assessing this balance provides insight into which mode of failure will occur first under increasing force.

Please give Ace some feedback