Question

Two masses A and B of $10\mathrm{kg}$ and $5\mathrm{kg}$ respectively, are connected with a string passing over a frictionless pulley at a corner of a table. The coefficient of friction of A with the table is $0.2$. Calculate the minimum mass of C that may be placed on A to prevent it from moving.

Answer 1

Step 1: Given data

Two masses A and B of $10\mathrm{kg}$ and $5\mathrm{kg}$.

Step 2: Concept used

1. In order to stay in equilibrium, the forces acting on the system must be in equilibrium.
2. The free-body diagram of the system can be drawn as follows:

3. The tension in the string is due to the weight of the mass B.

4. The tension must be in equilibrium with the frictional force exerted by the combined mass of A and mass C.

5. The weight of mass B can be calculated by multiplying the mass of B by the acceleration due to gravity.

6. Frictional force can be calculated by multiplying the combined weight of the mass A and mass C by the co-efficient of friction.

$$\begin{gathered} m_B\times g=\mu \left(m_A+m_C\right)g \\ 5\mathrm{kg}=0.2\left(10\mathrm{kg}+m_C\right) \end{gathered}$$

Where, $m_B$ is the mass of object B, $m_A$ is the mass of object A, $m_C$ is the mass of object C, $\mu$ is the co-efficient of friction, and $g$ is the acceleration due to gravity.

Step 3: Calculation of the mass C

Calculate the mass C as follows:

$$\begin{gathered} m_C=5\mathrm{kg}-2\mathrm{kg} \\ {m}_C=3\mathrm{kg} \end{gathered}$$

Hence, the mass of object C that needs to be placed is $3\mathrm{kg}$.