Two blocks with masses m1 = 1.10 kg and m2 = 3.10 kg are connected by a massless string, as shown in the Figure (the figure shows m1 on top of a box and m2 hanging off the side of the box). They are released from rest. The coefficent of kinetic friction between the upper block and the surface is 0.320.
Assume that the pulley has a negligible mass and is frictionless, and calculate the speed of the blocks after they have moved a distance 44.0 cm.
To calculate the speed of the blocks after they have moved a distance of 44.0 cm, we can apply the principles of Newton's laws and energy conservation.
Step 1: Determine the net force acting on the blocks.
The net force is the sum of the forces acting on the blocks. In this case, there are two forces to consider:
- The gravitational force on m1: F1 = m1 * g (where g is the acceleration due to gravity, approximately 9.8 m/s^2)
- The tension force due to the hanging mass m2: F2 = m2 * g
Step 2: Determine the frictional force.
The frictional force can be calculated using the coefficient of kinetic friction and the normal force. The normal force is the weight of m1, which is equal to m1 * g. Therefore, the frictional force is:
F_friction = μ * (m1 * g)
Step 3: Calculate the acceleration of the blocks.
The net force acting on the blocks is the difference between the tension force and the frictional force:
Net Force = F2 - F_friction
The acceleration of the blocks can be calculated using Newton's second law:
Acceleration = Net Force / (m1 + m2)
Step 4: Determine the work done by the net force.
The work done by the net force is equal to the force multiplied by the displacement:
Work = Net Force * displacement
Step 5: Apply the work-energy theorem.
According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. Therefore, the work done by the net force is equal to the change in kinetic energy:
Work = ΔKE
Step 6: Calculate the final velocity of the blocks.
The change in kinetic energy (ΔKE) is equal to the final kinetic energy minus the initial kinetic energy. Since the blocks start from rest, the initial kinetic energy is zero. Therefore:
Work = ΔKE = (1/2) * (m1 + m2) * v^2
Solve the equation for v:
v = sqrt((2 * Work) / (m1 + m2))
Given:
- Mass of m1 (m1) = 1.10 kg
- Mass of m2 (m2) = 3.10 kg
- Coefficient of kinetic friction (μ) = 0.320
- Distance (displacement) = 44.0 cm = 0.44 m
Now, let's calculate the speed of the blocks:
Step 1: Determine the net force acting on the blocks.
F1 = m1 * g = 1.10 kg * 9.8 m/s^2 = 10.78 N
F2 = m2 * g = 3.10 kg * 9.8 m/s^2 = 30.38 N
Step 2: Determine the frictional force.
F_friction = μ * (m1 * g) = 0.320 * (1.10 kg * 9.8 m/s^2) = 3.44 N
Step 3: Calculate the acceleration of the blocks.
Net Force = F2 - F_friction = 30.38 N - 3.44 N = 26.94 N
Acceleration = Net Force / (m1 + m2) = 26.94 N / (1.10 kg + 3.10 kg) = 5.39 m/s^2
Step 4: Determine the work done by the net force.
Work = Net Force * displacement = 26.94 N * 0.44 m = 11.86 J
Step 5: Apply the work-energy theorem.
Work = ΔKE = (1/2) * (m1 + m2) * v^2
Step 6: Calculate the final velocity of the blocks.
v = sqrt((2 * Work) / (m1 + m2)) = sqrt((2 * 11.86 J) / (1.10 kg + 3.10 kg)) ≈ 2.65 m/s
Therefore, the speed of the blocks after they have moved a distance of 44.0 cm is approximately 2.65 m/s.