Question

a person exerts a force ( f_p ) on a rope to pull a block of mass ( m ) up a ramp. the rope makes an angle ( \theta_2 ) to the ramp and the ramp makes an angle ( \theta_1 ) to the horizontal, as shown. the force of kinetic friction between the block and the ramp has magnitude ( f_f ). which of the following expressions is equal to the magnitude of the acceleration ( a ) of the block?

a ( \frac{f_p cos \theta_2 + f_f}{m} - g sin \theta_1 )
b ( \frac{f_p cos \theta_2 - f_f}{m} - g sin \theta_1 )
c ( \frac{f_p cos \theta_2 + f_f}{m} + g sin \theta_1 )
d ( \frac{f_p cos \theta_2 - f_f}{m} + g sin \theta_1 )

Explanation:

Step1: Analyze forces along the ramp

According to Newton's second law \(F = ma\), the net force along the ramp is \(F_{net}=F_p\cos\theta_2 - F_f - mg\sin\theta_1\).

Step2: Solve for acceleration \(a\)

Since \(F_{net}=ma\), then \(a=\frac{F_p\cos\theta_2 - F_f - mg\sin\theta_1}{m}=\frac{F_p\cos\theta_2 - F_f}{m}-g\sin\theta_1\)

Answer:

B. \(\frac{F_p\cos\theta_2 - F_f}{m}-g\sin\theta_1\)