Two blocks, of weights 2.8 N and 5.5 N, are connected by a massless string and slide down a 47° inclined plane. The coefficient of kinetic friction between the lighter block and the plane is 0.087; that between the heavier block and the plane is 0.39. Assuming that the lighter block leads, find (a) the magnitude of the acceleration of the blocks and (b) the tension in the string.
To find the magnitude of the acceleration of the blocks and the tension in the string, we can follow a step-by-step process:
Step 1: Calculate the net force acting on the system of blocks. The net force is the force that causes the acceleration of the blocks. We can determine the net force by considering the forces acting on each block separately.
For the lighter block:
- Weight (mg = mass x gravity) acts downwards with a magnitude of 2.8 N.
- The force of kinetic friction (fk = coefficient of kinetic friction x normal force) acts upwards.
- The component of the weight acting parallel to the incline (mg sinθ) acts downwards.
- The tension in the string (T) acts upwards.
The net force on the lighter block can be determined using the equation:
Net Force = T - fk - mg sinθ
For the heavier block:
- Weight (mg = mass x gravity) acts downwards with a magnitude of 5.5 N.
- The force of kinetic friction (fk = coefficient of kinetic friction x normal force) acts upwards.
- The component of the weight acting parallel to the incline (mg sinθ) acts downwards.
The net force on the heavier block is given by:
Net Force = -fk - mg sinθ
Step 2: Determine the acceleration of the blocks. Since the blocks are connected by a massless string, they have the same acceleration. We can set the net force equations for both blocks equal to each other (since they have the same acceleration), and solve for the acceleration (a).
T - fk - mg sinθ = -fk - mg sinθ + ma
T = ma
Step 3: Calculate the tension in the string. Now that we have the acceleration value (a), we can substitute it back into the equation for the tension in the string to solve for T.
T = ma
Now that we understand the steps, let's calculate the magnitude of the acceleration of the blocks and the tension in the string:
Given data:
Weight of lighter block (mg1) = 2.8 N
Weight of heavier block (mg2) = 5.5 N
Coefficient of kinetic friction between lighter block and the incline (μ1) = 0.087
Coefficient of kinetic friction between heavier block and the incline (μ2) = 0.39
Angle of the incline (θ) = 47°
Step 1:
Net Force on the lighter block:
Net Force1 = T - fk1 - mg1 sinθ
Net Force on the heavier block:
Net Force2 = -fk2 - mg2 sinθ
Step 2:
Net Force1 = Net Force2
T - fk1 - mg1 sinθ = -fk2 - mg2 sinθ + ma
Step 3:
T = ma
Now, we can substitute the given values into the equations and solve for the acceleration (a) and the tension in the string (T).