Two masses are connected by a string which passes over a pulley with negligible mass and friction. One mass hangs vertically and one mass slides on a 30.0 degree incline. The inclined surface has a coefficient of kinetic friction of 0.200. The vertically hanging mass is 4.00 kg and the mass on the incline is 6.00 kg. The magnitude of the acceleration of the 4.00 kg mass is (the initial velocity of the 6.00 kg mass is down the incline) A) 7.16 m/s². B) 5.84 m/s². C) 2.98 m/s². D) 1.53 m/s². E) 3.90 m/s².

Question

Quizplus · Accepted Answer

We can begin the problem by analyzing each mass separately. 
For the 6.00 kg mass on the incline, the force of gravity (mg) is resolved into two components: mgcos(30°) perpendicular to the incline and mgsin(30°) parallel to the incline. The force of friction (f) is equal to the coefficient of kinetic friction (μ) times the perpendicular component of the weight (mgcos(30°)). Therefore, since the 6.00 kg mass is moving downward, the net force on this mass is: 
Fnet = mgsin(30°) - μmgcos(30°)

For the 4.00 kg mass hanging vertically, the force of gravity (mg) is the only force acting on it since it is not moving laterally.

Since the two masses are connected by the same string, the tension force (T) in the string is the same for both masses. We can use this to relate the acceleration (a) of the 4.00 kg mass to the net force on the 6.00 kg mass: 
T - mg = ma 
T - mgsin(30°) + μmgcos(30°) = (m/2)a

Solving for T in the first equation and substituting into the second equation: 
mg - ma = mgsin(30°) - μmgcos(30°) 
a = g(sin(30°) - μcos(30°)) 
a = 5.84 m/s^2

Therefore, the answer is (B).

Two Masses Are Connected by a String Which Passes Over