Source: Principles of Physics
Problem
Two blocks of mass $m_1$ and $m_2$ are connected by a massless string. They slide down a plane inclined at an angle $\alpha$ to the horizontal. The coefficients of kinetic friction between the blocks and the plane are $\mu_1$ and $\mu_2$ respectively. Block $m_1$ is positioned downslope from block $m_2$.
- Find the magnitude of the blocks' acceleration, $a$.
- Find the tension in the string, $T$, assuming it is taut.
[Q1] $a = g \left( \sin\alpha - \frac{\mu_1 m_1 + \mu_2 m_2}{m_1+m_2} \cos\alpha \right)$ [Q2] $T = \frac{m_1 m_2 g (\mu_2 - \mu_1)}{m_1+m_2} \cos\alpha$
Let the positive direction be down the incline. We first analyze the entire system of total mass $M = m_1+m_2$. The net force along the incline is the sum of the gravitational components minus the total friction force. Tension is an internal force and does not affect the system's overall acceleration.
Applying Newton's second law to the system:
$$F_{net} = (m_1+m_2)g\sin\alpha - (\mu_1 m_1 g\cos\alpha + \mu_2 m_2 g\cos\alpha) = (m_1+m_2)a$$Solving for acceleration $a$:
$$a = g\sin\alpha - g\cos\alpha \left( \frac{\mu_1 m_1 + \mu_2 m_2}{m_1+m_2} \right)$$To find the tension $T$, we apply Newton's second law to the trailing block, $m_2$. The forces on it along the incline are its gravitational component, friction, and tension.
$$F_{2,net} = m_2g\sin\alpha + T - \mu_2 m_2g\cos\alpha = m_2a$$Isolating $T$ and substituting the expression for $a$:
$$T = m_2a - m_2g\sin\alpha + \mu_2 m_2g\cos\alpha$$ $$T = m_2 \left[ g\sin\alpha - g\cos\alpha \left( \frac{\mu_1 m_1 + \mu_2 m_2}{m_1+m_2} \right) \right] - m_2g\sin\alpha + \mu_2 m_2g\cos\alpha$$ $$T = g\cos\alpha \left[ \mu_2 m_2 - m_2 \left( \frac{\mu_1 m_1 + \mu_2 m_2}{m_1+m_2} \right) \right]$$ $$T = m_2 g\cos\alpha \left[ \frac{\mu_2(m_1+m_2) - (\mu_1 m_1 + \mu_2 m_2)}{m_1+m_2} \right]$$ $$T = \frac{m_2 g\cos\alpha (\mu_2 m_1 + \mu_2 m_2 - \mu_1 m_1 - \mu_2 m_2)}{m_1+m_2} = \frac{m_1 m_2 g \cos\alpha (\mu_2 - \mu_1)}{m_1+m_2}$$Note that for the string to be taut ($T > 0$), we must have $\mu_2 > \mu_1$.