block a 10 kg is connected to block b 20 kg by a mass less string on a horizontal surface (coefficient of kinetic friction 0.1). block b is connected by another string to a hanging mass M through a mass less pulley. what is the minimun value of M that will make the system move?
To determine the minimum value of M that will make the system move, we need to consider the forces acting on the system and the condition required for motion to occur.
First, let's analyze the forces acting on each block:
Block A (10 kg):
- Weight (W_A) = mass_A * gravity = 10 kg * 9.8 m/s^2 = 98 N (downward)
- Friction force (F_friction_A) = coefficient_of_friction * Normal force_A
= 0.1 * Normal force_A
Block B (20 kg):
- Weight (W_B) = mass_B * gravity = 20 kg * 9.8 m/s^2 = 196 N (downward)
- Tension force from the string connecting A and B (T_AB) = T (let's assume the tension in the string is T)
- Friction force (F_friction_B) = coefficient_of_friction * Normal force_B
= 0.1 * Normal force_B
- Tension force from the string connecting B and the hanging mass (T_BM) = T
Since the blocks are connected by a massless string, the tension force is the same on both sides of the string.
Now, let's consider the conditions needed for motion to occur:
To overcome static friction and initiate motion, the force applied to the system (T) should be greater than or equal to the maximum static friction force (F_friction_max). Once the block starts moving, it experiences kinetic friction.
Therefore, we need to compare the maximum static friction force with the force applied by the hanging mass M (T_BM) to find the minimum value of M that will make the system move.
The maximum static friction force can be calculated as:
F_friction_max = coefficient_of_friction * Normal force
For block A:
F_friction_max_A = 0.1 * Normal force_A
For block B:
F_friction_max_B = 0.1 * Normal force_B
Next, we need to find the normal forces on each block. The normal force acting on a block is equal to the weight of the block if it is on a horizontal surface:
Normal force_A = W_A = 98 N
Normal force_B = W_B = 196 N
Now, let's calculate the maximum static friction forces:
F_friction_max_A = 0.1 * 98 N = 9.8 N
F_friction_max_B = 0.1 * 196 N = 19.6 N
Since the tension force T applied by the hanging mass (M) is shared between both blocks, we have:
T = T_BM = T_AB
To find the minimum value of M that will make the system move, we need to ensure that T (force applied by M) is greater than or equal to the maximum static friction force acting on either block.
For block A: T >= F_friction_max_A
For block B: T >= F_friction_max_B
Substituting the values:
T >= 9.8 N (equation 1)
T >= 19.6 N (equation 2)
To find the minimum value of M, we should use the larger value obtained from equation 1 and equation 2. Thus, the minimum value of M that will make the system move is the force equivalent to the larger static friction force, which is 19.6 N.
Therefore, the minimum value of M required to make the system move is such that the hanging mass (M) exerts a force greater than or equal to 19.6 N.