Answer and Explanation:
a. To calculate the work done by the applied force, we can use the formula:
Work = Force * Distance * cos(theta)
where the force is 36 N, the distance is 2.4 m, and theta is 30° (converted to radians).
Plugging in the values, we have:
Work = 36 N * 2.4 m * cos(30°)
Using the value of cos(30°) ≈ 0.866, we can calculate:
Work = 36 N * 2.4 m * 0.866
Work ≈ 74.0 J
b. To calculate the work done by the force of gravity, we need to find the vertical displacement of the block. Since the block moves horizontally, the vertical displacement is 0. Therefore, the work done by the force of gravity is 0 J.
c. The work done by the normal force exerted by the table is also 0 J because the normal force acts perpendicular to the displacement of the block, resulting in no work done.
d. The work done by the force of friction can be calculated using the formula:
Work = Force of friction * Distance * cos(180°)
The force of friction can be found using the formula:
Force of friction = coefficient of friction * Normal force
Given that the coefficient of friction (μr) is 0.40 and the normal force is equal to the weight of the block (mass * gravitational acceleration), we have:
Force of friction = 0.40 * (4.0 kg * 9.8 m/s^2)
Now, we can calculate the work done by the force of friction:
Work = Force of friction * Distance * cos(180°)
Work = (0.40 * 4.0 kg * 9.8 m/s^2) * 2.4 m * cos(180°)
Work ≈ 37.6 J
e. The final kinetic energy of the block can be calculated using the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.
Since the block starts from rest, the initial kinetic energy is 0. Therefore, the work done by the applied force (74.0 J) is equal to the final kinetic energy.
f. To calculate the final speed of the block, we can use the equation:
Final Kinetic Energy = (1/2) * Mass * Final Velocity^2
Plugging in the values, we have:
74.0 J = (1/2) * 4.0 kg * Final Velocity^2
Simplifying the equation, we get:
74.0 J = 2.0 kg * Final Velocity^2
Dividing both sides by 2.0 kg, we have:
37.0 J/kg = Final Velocity^2
Taking the square root of both sides, we get:
Final Velocity ≈ 6.08 m/s
Therefore, the final speed of the block is approximately 6.08 m/s.