Solving a Two-Block System with Friction and Inclined Plane

Q1. What must be the mass $m_2$ of the hanging block if it is to descend 12.0 m in the first 3.00 s after the system is released from rest? (Given: $m_1 = 20.0$ kg, $\alpha = 53.1^\circ$, $\mu_k = 0.40$)

Background

Topic: Newton's Laws of Motion, Friction, and Kinematics

This question tests your ability to analyze a system of two connected masses, one on an inclined plane with friction and one hanging vertically, using Newton's second law and kinematic equations.

Key Terms and Formulas

• Newton's Second Law: $\sum \vec{F} = m \vec{a}$

• Kinetic Friction: $f_k = \mu_k N$

• Kinematic Equation for distance: $d = \frac{1}{2} a t^2$ (for motion starting from rest)

• Normal Force on Incline: $N = m_1 g \cos \alpha$

• Force components along incline: $m_1 g \sin \alpha$ (down the incline)

Step-by-Step Guidance

1. Start by identifying the forces acting on each block. For $m_1$ (on the incline): gravity (split into parallel and perpendicular components), friction, and tension. For $m_2$ (hanging): gravity and tension.

2. Write Newton's second law for each block. For $m_2$: $m_2 a = m_2 g - T$. For $m_1$: $m_1 a = T - \mu_k N - m_1 g \sin \alpha$.

3. Find the normal force for $m_1$: $N = m_1 g \cos \alpha$.

4. Use the kinematic equation to find the acceleration $a$ needed for $m_2$ to descend 12.0 m in 3.00 s: $a = \frac{2d}{t^2}$.

5. Substitute the value of $a$ into the equations for tension and solve for $T$ using the equation for $m_1$.

Try solving on your own before revealing the answer!