A horizontal force of magnitude F = 110 N is used to push a box of mass m = 6 kg from rest a distance d = 16 m up a frictionless incline with a slope q = 28°.
a) How much work is done by the force on the box?
b) How much work is done on the box by the gravitational force during this same displacement?
c) How much work is done by the normal force of the slope on the box during this displacement?
d) How fast is the box moving after this displacement?
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a) To determine the work done by the applied force, you can use the formula:
Work = Force * Displacement * cos(angle)
In this case, the magnitude of the applied force is given as F = 110 N and the displacement is d = 16 m. The angle between the force and displacement vectors is the angle of the incline, which is given as q = 28°. Plugging in these values, we have:
Work = 110 N * 16 m * cos(28°)
Simplifying, we find:
Work = 110 N * 16 m * cos(28°) ≈ 1647 J
Therefore, the work done by the force is approximately 1647 Joules.
b) The work done by the gravitational force can be calculated using the formula:
Work = Force * Displacement * cos(angle)
Here, the force due to gravity is given by the weight of the box, which can be calculated by multiplying the mass of the box (m) by the acceleration due to gravity (g). The displacement and angle remain the same as in part (a). Thus, we get:
Work = (mass * gravity) * Displacement * cos(angle)
= (6 kg * 9.8 m/s^2) * 16 m * cos(28°)
Evaluating this expression gives:
Work = (6 kg * 9.8 m/s^2) * 16 m * cos(28°) ≈ 1589 J
Therefore, the work done by the gravitational force is approximately 1589 Joules.
c) The normal force of the slope does not do any work since the direction of the force is perpendicular to the displacement.
Therefore, the work done by the normal force is zero.
d) To find the final velocity of the box after the displacement, you can use the work-energy theorem, which states that the work done on an object equals its change in kinetic energy. In this case, since the box starts from rest, the work done is equal to the change in kinetic energy:
Work = Change in Kinetic Energy
(110 N * 16 m * cos(28°)) = (1/2) * mass * velocity^2
Simplifying and solving for velocity:
velocity^2 = (2 * (110 N * 16 m * cos(28°))) / mass
velocity^2 ≈ 76 m^2/s^2
velocity ≈ √(76) m/s
velocity ≈ 8.72 m/s
Therefore, the final velocity of the box after this displacement is approximately 8.72 m/s.