A 55-kg block, starting from rest, is pushed a distance of 5.0m across a floor by a horizontal force Fp whose magnitude is 140N. Fp is parallel to the displacement of the block. The final speed of the block is 2.35m/s.
a. How much work was converted to thermal energy? What work did friction do on the box?
b. What is the coefficient of friction?
Let's break down the problem step by step:
a. To find the work converted to thermal energy, we need to calculate the work done by the force of friction.
The work done by a force can be calculated using the formula:
Work = Force * Distance * cos(theta),
where theta is the angle between the force and the displacement.
Since the force Fp is parallel to the displacement of the block, the angle theta is 0 degrees, and cos(theta) = 1.
Therefore, the work done by the force Fp is:
Work = Fp * Distance = 140N * 5.0m = 700J.
Now, let's calculate the work done by the force of friction. The work done by a force of friction can be calculated using the same formula:
Work_friction = Frictional Force * Distance * cos(theta),
where theta is the angle between the force of friction and the displacement.
Since the force of friction acts in the opposite direction to the displacement, the angle theta is 180 degrees, and cos(theta) = -1.
The net work done by all forces acting on the block is equal to the change in kinetic energy. Therefore, the work done by the force of friction is:
Work_friction = Change in kinetic energy = (1/2) * mass * (final velocity^2 - initial velocity^2) = (1/2) * 55kg * (2.35m/s)^2.
Now, we can calculate the work done by the force of friction:
Work_friction = (1/2) * 55kg * (2.35m/s)^2 = 166.53625J.
To find the work converted to thermal energy, we subtract the work done by the force of friction from the work done by the force Fp:
Work_thermal = Work - Work_friction = 700J - 166.53625J.
Therefore, the work converted to thermal energy is:
Work_thermal = 533.46375J.
b. To find the coefficient of friction, we use the formula:
Force_friction = coefficient of friction * normal force,
where the normal force is equal to the weight of the block.
Given that the mass of the block is 55kg, the weight can be calculated as:
Weight = mass * acceleration due to gravity = 55kg * 9.8m/s^2.
Since the force of friction is acting horizontally, the normal force is equal to the weight of the block.
Therefore, the force of friction can be calculated as:
Force_friction = 140N.
Using the formula above, we can rearrange it to find the coefficient of friction:
coefficient of friction = Force_friction / Weight.
Plugging in the values, we get:
coefficient of friction = 140N / (55kg * 9.8m/s^2).
Therefore, the coefficient of friction is:
coefficient of friction = 0.256.
To find the answers to these questions, we first need to understand the concept of work, friction, and the work-energy principle.
a. To calculate the amount of work that was converted to thermal energy and the work done by friction, we can use the work-energy principle. According to this principle, the net work done on an object is equal to the change in its kinetic energy. In this case, since the block starts from rest, the initial kinetic energy is zero.
The net work done on the block is given by the equation:
Net work = Change in Kinetic Energy
Change in Kinetic Energy = 1/2 * mass * (final velocity)^2 - 1/2 * mass * (initial velocity)^2
Since the block starts from rest, the initial velocity is zero, and the final velocity is given as 2.35 m/s. The mass of the block is 55 kg.
Using these values, we can calculate the change in kinetic energy:
Change in Kinetic Energy = 1/2 * 55 kg * (2.35 m/s)^2 - 1/2 * 55 kg * (0 m/s)^2
= 1/2 * 55 kg * (2.35^2) - 1/2 * 55 kg * (0^2)
= 1/2 * 55 kg * (5.5225) - 1/2 * 55 kg * (0)
= 1/2 * 55 kg * 5.5225
= 151.7125 J
Therefore, the change in kinetic energy, which represents the work done on the block, is 151.7125 J.
Now, since there is no other work being done on the block, the net work done on the block is equal to the work done by friction and the work converted to thermal energy. So, the amount of work converted to thermal energy and the work done by friction is 151.7125 J.
b. To find the coefficient of friction (μ), we can use the equation:
Frictional Force = Coefficient of Friction * Normal Force
where the normal force is the force exerted by the surface on the block, which is equal to the weight of the block (mass * gravitational acceleration).
The frictional force can be calculated using the equation:
Frictional Force = Applied Force - Net Force
From the given data, the applied force is 140 N, and the net force is the force required to accelerate the block, which can be calculated using Newton's second law:
Net Force = mass * acceleration
In this case, the block starts from rest and reaches a final speed, so we can use the equation:
Net Force = mass * (final velocity - initial velocity)/(time taken)
Since the block starts from rest, the initial velocity is zero, and we are not given the time taken, we cannot directly calculate the acceleration or net force. Therefore, we cannot determine the coefficient of friction without additional information.
Please provide any additional data or clarify the information given if you need a more accurate answer to part (b).