Consider the system shown, with m1 = 2.0 kg, m2 = 3.0 kg, and α = 30o. The force F acts horizontally to the left and the kinetic coefficient of friction between m1 and the incline is μk = 0.40. The mass m2 is moving up at a constant speed.

(a) What is the magnitude of F?
(b) If the mass m2 moves up a distance of 2.0 m,
what is the work done by the following forces
acting on m1: (i) the force F, (ii) the normal force
due to the incline, (iii) the tension, (iv) friction, and
(v) gravity? What is the net work done on m1 by all the forces?
Spring 2013
F
(c) If the mass m2 moves up a distance of 2.0 m, what is the work done by the following forces acting on m2: (i) the tension and (ii) gravity? What is the net work done on m2 by all the forces?

To solve this problem, we need to analyze the forces acting on the system and use the principles of Newtonian mechanics.

First, let's calculate the force F required to keep the system in equilibrium. From the given information, we know that mass m2 is moving up the incline at a constant speed. This means that the net force acting on m2 must be zero. Therefore, the force of gravity acting on m2 must be balanced by the tension in the string.

(a) To find the magnitude of F, we can set up a force balance equation for m2 in the vertical direction:

m2 * g - T = 0

where m2 is the mass of object 2 (3.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and T is the tension in the string. Solving for T, we have:

T = m2 * g = 3.0 kg * 9.8 m/s^2 = 29.4 N

Since the force F must be equal in magnitude to the tension T, the magnitude of F is also 29.4 N.

(b) To calculate the work done by each force acting on m1, we need to consider the displacement of m1 and the force applied in that direction. Note that the displacement is given as 2.0 m.

(i) The work done by the force F is given by the equation:

Work = F * displacement * cos(angle)

Since the force F acts horizontally to the left and the displacement is in the same direction, the angle between them is 0 degrees. Therefore, the work done by F is:

Work_F = F * displacement * cos(0) = F * displacement

(ii) The normal force does not do any work since it acts perpendicular to the displacement.

(iii) The tension also does not do any work since it acts perpendicular to the displacement.

(iv) The work done by friction can be calculated using the equation:

Work_friction = frictional force * displacement

The frictional force can be found using the equation:

frictional force = μk * normal force

where μk is the kinetic coefficient of friction between m1 and the incline. The normal force can be calculated using the equation:

normal force = m1 * g * cos(α)

Substituting the given values, we have:

normal force = 2.0 kg * 9.8 m/s^2 * cos(30 degrees) = 16.78 N

frictional force = 0.40 * 16.78 N = 6.712 N

Therefore, the work done by friction is:

Work_friction = 6.712 N * 2.0 m = 13.42 J

(v) The work done by gravity can be calculated using the equation:

Work_gravity = gravitational force * displacement

The gravitational force can be found using the equation:

gravitational force = m1 * g * sin(α)

Substituting the given values, we have:

gravitational force = 2.0 kg * 9.8 m/s^2 * sin(30 degrees) = 9.8 N

Therefore, the work done by gravity is:

Work_gravity = 9.8 N * 2.0 m = 19.6 J

To find the net work done on m1 by all the forces, we sum up the work done by each force:

Net work on m1 = Work_F + Work_friction + Work_gravity

(c) Similarly, to calculate the work done on m2, we consider the displacement of m2 and the forces acting on it: the tension and gravity.

(i) The work done by the tension is given by:

Work_tension = T * displacement * cos(angle)

Since the tension and the displacement are in the same direction, the angle between them is 0 degrees. Therefore, the work done by the tension is:

Work_tension = T * displacement * cos(0) = T * displacement

(ii) The work done by gravity is given by:

Work_gravity = gravitational force * displacement

The gravitational force acting on m2 is:

gravitational force = m2 * g

Substituting the given values, we have:

gravitational force = 3.0 kg * 9.8 m/s^2 = 29.4 N

Therefore, the work done by gravity is:

Work_gravity = 29.4 N * 2.0 m = 58.8 J

To find the net work done on m2 by all the forces, we sum up the work done by each force:

Net work on m2 = Work_tension + Work_gravity