A construction worker attempts to lift a uniform beam off...

A construction worker attempts to lift a uniform beam off the floor and raise it to a vertical position. The beam is 2.50 $\mathrm{m}$ long and weighs 500 $\mathrm{N} .$ At a certain instant the worker holds the beam momentarily at rest with one end at distance $d=$ 1.50 $\mathrm{m}$ above the floor, as shown in Fig. $12-75,$ by exerting a force $\vec{P}$ on the beam, perpendicular to the beam. (a) What is the magnitude $P ?$ (b) What is the magnitude of the (net) force of the floor on the beam? (c) What is the minimum value the coefficient of static friction between beam and floor can have in order for the beam not to slip at this instant?

A construction worker attempts to lift a uniform beam off the
floor and raise it to a vertical position. The beam is 2.50 $\mathrm{m}$ long and
weighs 500 $\mathrm{N} .$ At a certain instant the worker holds the beam momentarily
at rest with one end at distance $d=$ 1.50 $\mathrm{m}$ above the floor, as shown in
Fig. $12-75,$ by exerting a force $\vec{P}$ on the beam, perpendicular to the
beam. (a) What is the magnitude $P ?$ (b) What is the magnitude of the (net) force of the floor on the beam? (c) What is the minimum value the coefficient of static
friction between beam and floor can have in order for the beam not to slip at this instant?

Key Concepts

Center of Mass

For a uniform beam, the center of mass is located at its midpoint, where the entire weight of the beam can be considered to act. This simplification is pivotal in static problems as it allows the weight to be represented by a single force acting at a known distance from any chosen pivot point, thereby facilitating the calculation of torques.

Friction and Coefficient of Static Friction

Friction is the resisting force that occurs when two surfaces interact, and the coefficient of static friction quantifies the maximum possible static friction force relative to the normal force between the surfaces. In this problem, it is essential to determine the threshold at which the beam will not slip relative to the floor. The minimum coefficient of static friction required prevents the beam from moving, ensuring equilibrium under the applied forces.

Static Equilibrium

This concept involves a body being at rest, meaning that the net force and net torque acting on the body are both zero. In the context of the problem, it ensures that the sum of the forces in all directions and the sum of the moments (torques) about any pivot point must cancel out. This is fundamental when analyzing forces and torques acting on a beam, such as gravitational forces, applied forces, and frictional forces, to ensure the beam remains momentarily stationary.

Torque (Moment of a Force)

Torque is the rotational equivalent of force and is calculated as the product of a force and its lever arm, which is the perpendicular distance from the pivot point to the line of action of the force. In the problem, torque balance is used to determine unknown forces. By setting the sum of torques about a specific point to zero, one can relate the applied force and the weight of the beam acting at the center of mass.

Transcript

Please give Ace some feedback