Consider the system shown, with m1 = 2.0 kg,
m2 = 3.0 kg, and á = 30o. The force F acts
horizontally to the left and the kinetic coefficient of
friction between m1 and the incline is ìk = 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?
(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?
The system shown?
To solve this problem, we need to analyze the forces acting on both masses and use the principles of Newtonian mechanics, specifically the work-energy theorem.
Let's start by calculating the magnitude of force F in part (a):
Step 1: Determine the vertical and horizontal components of the force F.
Since the angle is given as 30 degrees, we can calculate the horizontal component of F by multiplying the magnitude of F by the cosine of the angle: Fx = F * cos(30°).
Step 2: Calculate the frictional force.
The kinetic coefficient of friction (μk) between m1 and the incline is given as 0.40. The frictional force (Ff) can be found using the equation: Ff = μk * (normal force of m1).
Step 3: Determine the net force acting on m1 in the horizontal direction.
The net force (Fnet) acting on m1 is the difference between the horizontal component of force F and the frictional force: Fnet = Fx - Ff.
Step 4: Consider the equilibrium of forces in the vertical direction.
Since mass m2 is moving up at a constant speed, the net force acting on it in the vertical direction is zero. Hence, the vertical component of the force F must balance the weight of m2.
Now let's move on to part (b):
Step 1: Calculate the work done by force F.
The work done by a constant force is given by the equation: Work = Force * Distance * cos(θ), where θ is the angle between the force and the displacement. Here, since the force F is applied horizontally, the angle θ will be 0. We can use the magnitude of force F calculated in part (a) and multiply it by the horizontal displacement (2.0 m) to find the work done by force F on m1.
Step 2: Find the work done by the normal force.
The normal force does not do any work since it acts perpendicular to the displacement. Therefore, the work done by the normal force is zero.
Step 3: Calculate the work done by the tension force.
The tension force acts along the displacement, so the angle between the tension and displacement is 0. We can find the work done by the tension force by multiplying its magnitude by the displacement of m1.
Step 4: Determine the work done by friction.
The work done by friction can be calculated using the equation: Work = Frictional Force * Distance * cos(180°), since the force of friction acts opposite to the displacement.
Step 5: Calculate the work done by gravity.
The work done by gravity is given by the equation: Work = Force of Gravity * Distance * cos(θ), where θ is the angle between the force of gravity and the displacement. In this case, the angle θ is 180° since the displacement is in the opposite direction of the force of gravity.
Step 6: Calculate the net work done on m1.
The net work done on m1 is the sum of the work done by all forces acting on it.
Finally, let's move on to part (c):
Step 1: Calculate the work done by the tension force acting on m2.
Since the tension force acts along the displacement, the angle between the force and the displacement is 0. We can use the magnitude of the tension force and multiply it by the displacement of m2.
Step 2: Calculate the work done by gravity on m2.
The work done by gravity is given by the equation: Work = Force of Gravity * Distance * cos(θ). Here, the displacement and force of gravity are in opposite directions, so the angle θ is 180°.
Step 3: Calculate the net work done on m2.
The net work done on m2 is the sum of the work done by all forces acting on it.
By following these steps, you should be able to find the answers to all parts of the problem.