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 solve this problem, we can use the principles of Newton's laws of motion and the concept of work and energy.
First, let's analyze the forces acting on the system. We have the force of gravity acting on both blocks, and the tension in the string connecting the blocks. Additionally, there is kinetic friction between the upper block and the surface. Since the pulley is frictionless, we don't need to consider any forces related to it.
Let's break it down into steps:
1. Calculate the force of gravity on each block.
The force of gravity acting on an object can be calculated using the formula:
F_gravity = mass * acceleration_due_to_gravity
For the upper block (m1), the force of gravity is:
F_gravity1 = m1 * g
For the hanging block (m2), the force of gravity is:
F_gravity2 = m2 * g
Where g is the acceleration due to gravity, approximately 9.8 m/s².
2. Calculate the net force acting on the system.
The net force acting on the system can be determined by considering the forces involved. We have the tension in the string pulling m1 to the right and the force of kinetic friction opposing the motion:
Net force = Tension - Force of kinetic friction
Since the blocks are connected by a massless string, the tension throughout the string is constant. Therefore, the tension Tension is the same for both blocks.
3. Calculate the work done by kinetic friction.
The work done by kinetic friction can be calculated using the formula:
Work = Force * Distance * cos(θ)
For kinetic friction, the work done is negative because it opposes the motion, so:
Work_friction = -Force_friction * Distance
Here, Distance refers to the distance the upper block has moved.
4. Calculate the change in potential energy for the hanging block.
The change in potential energy for the hanging block (m2) is given by:
ΔPE2 = m2 * g * h
Here, h represents the height difference between the initial and final positions. Since both blocks move the same distance, h is equal to the distance the upper block has moved.
5. Apply the work-energy principle.
According to the work-energy principle, the total work done on an object is equal to the change in its kinetic energy:
Work_net = ΔKE
The net work done on the system is equal to the sum of the work done by friction (negative) and the change in potential energy:
Work_net = Work_friction + ΔPE2
6. Calculate the speed of the blocks.
The final step is to determine the speed of the system after the blocks have moved the given distance. We can use the equation for kinetic energy to find the speed:
KE = (1/2) * M * v²
Where M is the total mass of the system, given by M = m1 + m2, and v is the final velocity of the blocks.
By substituting the values we obtained in the previous steps, we can solve for v.