Question

1. Two masses hang on either side of a pulley. One mass (5.0 kg) is resting on a ramp with μ=0.25 and an incline of 30°. The second mass (8.0 kg) hangs freely.

a) Draw FBDs for both blocks.

b) Determine the acceleration of the block on the ramp. 

C) Use the hanging mass to determine the force of tension

Answer 1

(a) Let's draw FBDs for both blocks:

(b) Let's apply the Newton's Second Law of Motion:

$$F_T-m_1gsin\theta-\mu m_1gcos\theta=m_1a,$$ $$F_T-m_2g=-m_2a.$$

Solving the second equation for $F_T$ and substituting it into the first equation, we get:

$$-m_2a+m_2g-m_1gsin\theta-\mu m_1gcos\theta=m_1a,$$ $$a=\dfrac{m_2g-m_1g(sin\theta+\mu cos\theta)}{m_1+m_2},$$ $$a=\dfrac{8\ kg\cdot9.8\ \dfrac{m}{s^2}-5\ kg\cdot9.8\ \dfrac{m}{s^2}\cdot(sin30^{\circ}+0.25\cdot cos30^{\circ})}{5\ kg+8\ kg},$$ $$a=3.33\ \dfrac{m}{s^2}.$$

(c) We can find the force of tension from the second equation:

$$F_T=m_2(g-a)=8\ kg\cdot(9.8\ \dfrac{m}{s^2}-3.33\ \dfrac{m}{s^2})=51.76\ N.$$