Chapter 4, Problem 63P

Two masses m_A= 2.0 kg and m_B= 5.0 kg are on inclines and are connected together by a string as shown in Fig. 4-61. The coefficient of kinetic friction between each mass and its incline is ${\mu }_k$ = 0.30. If m_Amoves up, and m_B moves down, determine their acceleration. [Ignore masses of the (frictionless) pulley and the cord.]

Figure 4-61 Problem 65.

To determine

Two masses$m_a=2\text{kg}$ and $m_b=5\text{kg}$ are on inclines and are connected by a string. The coefficient of kinetic friction between each mass and its incline is ${\mu }_k=3.0$. If $m_a$ moves up, and $m_b$ moves down, determine their acceleration.

Answer to Problem 63P

Solution:

Acceleration = a = 2.16 m/s².

Explanation of Solution

Given:

$\begin{array}{l}{m}_a=2\text{kg}\\ m_b=5\text{kg}\\ {\mu }_k=3.0\end{array}$

Formula used:

${\displaystyle \sum F=ma}$

Calculation:

$\begin{array}{l}\text{For block a,}\\ T-m_{a}g\sin \theta -{\mu }_{k}mg\cos \theta =m_{a}a\\ T-2(9.8)\sin {51}^{\circ }-0.3(2)(9.8)\cos {51}^{\circ }=2a\\ T-18.93=2a----------(1)\\ \text{for block b}\\ m_{b}g\sin \theta -{\mu }_{k}mg\cos \theta -T=m_{b}a\\ 5(9.8)\sin {21}^{\circ }-0.3(5)(9.8)\cos {21}^{\circ }-T=5a\\ 3.84-T=5a----------(2)\\ \text{Adding equations (1) and (2), we get}\\ (3.84-T)+(T-18.93)=5a+2a\\ -15.09=7a\\ a=-2.16{\text{ m/s}}^2\end{array}$