A box of mass 250 kg slides 4.6 m down an inclined plane of 31°. Jim pushes against the box to keep it from accelerating. The coefficient of kinetic friction between the box and the ramp is 0.41
(a) Calculate the magnitude of the force exerted by Jim.
(b) Calculate the work done by Jim on the box.
(c) Calculate the work done by the force of friction.
(d) What is the work done by the force of gravity?
(e) What is the net work done on the box?
To solve this problem, we will use the following equations:
(a) The force exerted by Jim:
The force exerted by Jim can be determined using the equation:
Force exerted by Jim = Net force on the box = m * a
Here, m is the mass of the box and a is the acceleration of the box. Since the box is not accelerating, the net force on the box is zero. Therefore, the force exerted by Jim is also zero.
(b) The work done by Jim:
The work done by Jim can be calculated using the equation:
Work done = Force exerted by Jim * distance
Since the force exerted by Jim is zero, the work done by Jim is also zero.
(c) The work done by the force of friction:
The work done by the force of friction can be calculated using the equation:
Work done by friction = Force of friction * distance
To find the force of friction, we need to use the equation:
Force of friction = coefficient of friction * normal force
The normal force is equal to the weight of the box, which can be calculated using the equation:
Weight of the box = m * g
Here, g is the acceleration due to gravity.
(d) The work done by the force of gravity:
The work done by the force of gravity can be calculated using the equation:
Work done by gravity = weight of the box * distance
(e) The net work done on the box is equal to the sum of the work done by Jim, the work done by the force of friction, and the work done by the force of gravity.
Let's calculate the values step-by-step:
(a) The force exerted by Jim:
Force exerted by Jim = 0 N
(b) The work done by Jim:
Work done by Jim = 0 J
(c) The work done by the force of friction:
Force of friction = coefficient of friction * normal force
Normal force = weight of the box = m * g
Force of friction = 0.41 * (250 kg * 9.8 m/s^2) = 1009.9 N
Work done by friction = Force of friction * distance = 1009.9 N * 4.6 m = 4646 J
(d) The work done by the force of gravity:
Weight of the box = m * g = 250 kg * 9.8 m/s^2 = 2450 N
Work done by gravity = Weight of the box * distance = 2450 N * 4.6 m = 11270 J
(e) The net work done on the box:
Net work done = Work done by Jim + Work done by friction + Work done by gravity
Net work done = 0 J + 4646 J + 11270 J = 15916 J
Therefore:
(a) The magnitude of the force exerted by Jim is 0 N.
(b) The work done by Jim is 0 J.
(c) The work done by the force of friction is 4646 J.
(d) The work done by the force of gravity is 11270 J.
(e) The net work done on the box is 15916 J.
To calculate the force exerted by Jim, we need to consider the forces acting on the box. There are four significant forces: the force of gravity (mg), the normal force (N), the force exerted by Jim (F), and the force of friction (f). Since the box is not accelerating vertically, the normal force (N) cancels out the force of gravity (mg) in the vertical direction, leaving only the force of gravity acting parallel to the inclined plane.
The force exerted by Jim should equal the force of friction to prevent the box from accelerating. The equation for the force of friction is given by:
f = coefficient of friction * N
where N is the normal force. To calculate N, we use the fact that the normal force is equal in magnitude and opposite in direction to the vertical component of the weight of the box:
N = mg * cos(theta)
where theta is the angle of the inclined plane with respect to the horizontal. Plugging in the values, we have:
N = (250 kg) * (9.8 m/s^2) * cos(31°)
Now, we calculate the force of friction:
f = (0.41) * (250 kg) * (9.8 m/s^2) * cos(31°)
(a) Magnitude of the force exerted by Jim = force of friction = f
To calculate the work done by Jim on the box, we use the formula:
Work = force * distance
In this case, the force exerted by Jim is constant and equal to the force of friction, f. The distance is given as 4.6 m. Plugging in the values, we have:
(b) Work done by Jim = f * d = (force of friction) * (4.6 m)
To calculate the work done by the force of friction, we use the same formula:
(c) Work done by the force of friction = force of friction * distance = f * d
To calculate the work done by the force of gravity, we need to consider the displacement of the box along the inclined plane. The work done by gravity is given by:
Work = force * distance * cos(theta)
In this case, the force of gravity acting along the inclined plane is mg * sin(theta), and the distance is 4.6 m. Plugging in the values, we have:
(d) Work done by the force of gravity = (mg * sin(theta)) * d
To calculate the net work done on the box, we sum up the work done by all the forces involved. We have:
(e) Net work done on the box = Work done by Jim + Work done by the force of friction + Work done by the force of gravity
Now, you can substitute the values for mass, angle, distance, and other given quantities into the respective formulas to calculate the answers.