Question 4.SP.11: A 100-lb force acts as shown on a 300-lb crate placed on an ...
A 100-lb force acts as shown on a 300-lb crate placed on an inclined plane. The coefficients of friction between the crate and the plane are \mu_s = 0.25 and \mu_k = 0.20. Determine whether the crate is in equilibrium, and find the value of the friction force.
STRATEGY: This is a friction problem of the first type: You know the forces and the friction coefficients and want to determine if the crate moves. You also want to find the friction force.
MODELING and ANALYSIS:
Force Required for Equilibrium. First determine the value of the friction force required to maintain equilibrium. Assuming that F is directed down and to the left, draw the free-body diagram of the crate (Fig. 1) and solve the equilibrium equations:
\begin{aligned}&+\nearrow \Sigma F_x=0: \quad 100 lb-\frac{3}{5}(300 lb)-F=0 \\&F=-80 lb \quad \pmb{F}=80 lb \nearrow \\&+\nwarrow \Sigma F_y=0: \quad N-\frac{4}{5}(300 lb)=0 \\&N=+240 lb \quad \pmb{N}=240 lb \nwarrow \end{aligned}The force F required to maintain equilibrium is an 80-lb force directed up and to the right; the tendency of the crate is thus to move down the plane.
Maximum Friction Force. The magnitude of the maximum friction force that may be developed between the crate and the plane is
F_m=\mu_s N \quad \quad F_m=0.25(240 lb)=60 lbBecause the value of the force required to maintain equilibrium (80 lb) is larger than the naximum value that may be obtained (60 lb), equilibrium is not maintained and the crate will side down the plane.
Actual Value of Friction Force. The magnitude of the actual friction force is
F_{\text {actual }}=F_k=\mu_k N=0.20(240 lb)=48 lbThe sense of this force is opposite to the sense of motion; the force is thus directed up and to the ight (Fig. 2):
\pmb{F}_{\text {actual }}=48 lb \nearrowNote that the forces acting on the crate are not balanced. Their resultant is
\frac{3}{5}(300 lb)-100 lb-48 lb=32 lb \swarrowREFLECT and THINK: This is a typical friction problem of the first type. Note that you used the coefficient of static friction to determine if the crate moves, but once you found that it does move, you needed the coefficient of kinetic friction to determine the friction force.
