a. To calculate the work done by gravitational force over the ground trip, we can use the formula:
Work = Force x Distance x cos(theta)
where the force is the weight of the body, which is given by:
Force = mass x acceleration due to gravity
Force = 0.5 kg x 9.81 m/s^2 (acceleration due to gravity)
Force = 4.905 N
The distance covered by the body is the length of the inclined plane, which is 10 meters. The angle between the direction of the force and the direction of motion is zero since the force and the motion are in the same direction. Therefore, the cosine of the angle is 1.
Work = 4.905 N x 10 m x 1
Work = 49.05 Joules
Therefore, the work done by gravitational force over the ground trip is 49.05 Joules.
b. Since the body is taken up the inclined plane, an applied force is needed to overcome the force of gravity. However, the problem statement does not mention any applied force, so we cannot calculate the work done by an applied force.
c. To calculate the work done by the frictional force over the round trip, we can use the formula:
Work = Force x Distance x cos(theta)
where the force is the frictional force, which is given by:
Force = coefficient of friction x normal force
The normal force is equal to the weight of the body, which we already calculated as 4.905 N. Therefore:
Force = 0.12 x 4.905 N
Force = 0.5886 N
The distance covered by the body is twice the height of the inclined plane, or 16 meters. The angle between the direction of the force and the direction of motion is 180 degrees since the force and the motion are in opposite directions. Therefore, the cosine of the angle is -1.
Work = 0.5886 N x 16 m x (-1)
Work = -9.4176 Joules
Note that the negative sign indicates that the work done by the frictional force is in the opposite direction to the motion of the body.
Therefore, the work done by the frictional force over the round trip is -9.4176 Joules.
d. To calculate the kinetic energy of the body at the end of the trip, we can use the formula:
Kinetic energy = 0.5 x mass x velocity^2
At the top of the inclined plane, the body is at rest, so its initial velocity is zero. At the bottom of the inclined plane, the body has gained kinetic energy due to the work done by gravitational force. We can calculate the final velocity using the principle of conservation of energy:
Work done by gravitational force = Change in kinetic energy
The work done by gravitational force is equal to the work we calculated in part (a), or 49.05 Joules. Therefore:
49.05 Joules = 0.5 x 0.5 kg x (final velocity)^2
(final velocity)^2 = 98.1
final velocity = 9.905 m/s
Substituting this value into the formula for kinetic energy, we get:
Kinetic energy = 0.5 x 0.5 kg x (9.905 m/s)^2
Kinetic energy = 24.652 Joules
Therefore, the kinetic energy of the body at the end of the trip is 24.652 Joules.