Q1. Two blocks are connected as shown: sits on top of , with a coefficient of kinetic friction between them. The floor is frictionless. A horizontal force is applied to . Find: (a) the tension in the string, and (b) the acceleration of block $m_2$.

Background

Topic: Newton's Laws of Motion, Friction, and Dynamics of Connected Objects

This problem tests your ability to analyze forces in a system of two blocks with friction between them, but no friction with the floor. You must apply Newton's Second Law to each block, account for the frictional force, and use the constraint imposed by the string.

Key Terms and Formulas

• Newton's Second Law:

• Kinetic Friction:

• Tension (): The force transmitted by the string between the blocks.

• Normal Force (): The perpendicular contact force between two surfaces.

Step-by-Step Guidance

1. Draw free-body diagrams for both blocks. Identify all forces acting on (top block) and (bottom block): gravity, normal force, friction, tension, and the applied force .

   

2. Write Newton's Second Law for (top block): where is the kinetic friction force between $m_1$ and .

3. Write Newton's Second Law for (bottom block): Note: is the friction force exerted by on $m_2$ (equal in magnitude to by Newton's Third Law).

4. Express the friction force using , where is the normal force between and . Since the only vertical forces on $m_1$ are gravity and the normal force from $m_2$, .

5. Use the constraint that the string keeps the blocks moving together horizontally (if not slipping): .

6. Combine the equations to solve for and . Set up the system of equations, but stop before plugging in the numbers for the final calculation.

Try solving on your own before revealing the answer!