An arrangement of 3 blocks A ,B,C such that block A of mass 5 kg is over block B of 6 kg, with a friction between them of 0.8. both the blocks are connected to the 3rd block C of mass m by a string and frictionless pulley. Now to find at what value of m will both pulley A and B move together?
To determine the value of mass (m) at which both block A and block B move together, we need to analyze the forces acting on each block and consider the principles of Newton's laws of motion.
Let's break down the forces acting on each individual block:
1. Block A (mass = 5 kg):
- Force of gravity acting downward (weight) = 5 kg * 9.8 m/s^2 ≈ 49 N.
- Normal force exerted by block B acting upward = 49 N (since both blocks are in contact and not moving vertically).
- Friction force opposing the motion between block A and B = 0.8 * Normal force = 0.8 * 49 N = 39.2 N.
2. Block B (mass = 6 kg):
- Force of gravity acting downward (weight) = 6 kg * 9.8 m/s^2 ≈ 58.8 N.
- Force exerted by block A acting downward = 58.8 N (due to the weight of block B pushing against block A).
- Friction force opposing the motion between block A and B = 39.2 N (same as for block A).
To ensure that both blocks A and B move together, we need the forces of friction to be greater than the force that tends to make them slide apart.
Thus, the maximum force of friction between block A and B can be given by min(Friction A, Friction B) = min(39.2 N, 39.2 N) = 39.2 N.
Now, let's consider block C (mass = m). Since it is connected to both block A and block B by a string and a frictionless pulley, the tension in the string will be the same for both blocks.
The tension force is given by the force required to lift block C and is equal to the weight of block C, which is m * 9.8 m/s^2.
To have both blocks A and B move together, the maximum force of friction (39.2 N) must be equal to or greater than the tension force (m * 9.8 m/s^2).
Therefore, the condition for both blocks to move together is:
39.2 N ≥ m * 9.8 m/s^2
By rearranging the equation, we can find the critical mass value (m):
m ≤ 39.2 N / 9.8 m/s^2 ≈ 4 kg
Therefore, both blocks A and B will move together as long as the mass of block C (m) is less than or equal to 4 kg.