Problem 3- A block of mass m1 = 4 kg moves from rest under the influence of a varying force F(x) = 4x + 38, where F is in newton and x is in meters, on a rough horizontal surface having a coefficient of kinetic friction of 0.2. The block is connected by a light cord passing over a small frictionless pulley to a second hanging block of mass m2 = 3 kg as shown. If the system moves from x= 0 to x= 2m, calculate:
1) The work done by the force F(x) to move m1 from x= 0 to x= 2 m.
2) The change in the potential energy of each mass
3) The work done by friction.
4) The change in the kinetic energy of the system.
To solve this problem, we need to break it down into different components and calculate them one by one. Let's go through each part step by step:
1) The work done by the force F(x) to move m1 from x=0 to x=2m:
To calculate the work done by a variable force, we can use the equation:
Work = ∫ F(x) dx
In this case, the force F(x) = 4x + 38. To find the work done by this force, we integrate it with respect to x from 0 to 2m:
Work = ∫ (4x + 38) dx, limits (0 to 2m)
Integrating this equation, we get:
Work = 2x^2 + 38x, limits (0 to 2m)
Substituting the limits:
Work = 2(2m)^2 + 38(2m) - (2(0)^2 + 38(0))
Work = 8m^2 + 76m
Therefore, the work done by the force F(x) to move m1 from x=0 to x=2m is 8m^2 + 76m Joules.
2) The change in the potential energy of each mass:
The potential energy of an object is given by the equation:
Potential Energy = mass * acceleration due to gravity * height
For the first mass m1, the height can be calculated using the equation x = height. Hence, the potential energy change for m1 is:
Potential Energy (m1) = m1 * g * (x final - x initial)
Potential Energy (m1) = 4kg * 9.8m/s^2 * (2m - 0)
Potential Energy (m1) = 78.4 Joules
For the second mass m2, since it moves vertically downward, the height is given by the equation height = x initial - x final. Thus, the potential energy change for m2 is:
Potential Energy (m2) = m2 * g * (x initial - x final)
Potential Energy (m2) = 3kg * 9.8m/s^2 * (0 - 2m)
Potential Energy (m2) = -58.8 Joules
Therefore, the potential energy change for m1 is 78.4 Joules, and for m2 is -58.8 Joules.
3) The work done by friction:
The work done by friction can be calculated using the equation:
Work (friction) = force of friction * distance
The force of friction is given by the equation:
Force of friction = coefficient of friction * normal force
The normal force is equal to the weight of the rectangular block with mass m1:
Normal force = m1 * g
Therefore, the force of friction can be calculated as:
Force of friction = coefficient of friction * (m1 * g)
The distance over which the force of friction acts is equal to the distance the block m1 moves, which is 2m. Therefore, we can calculate the work done by friction as:
Work (friction) = coefficient of friction * (m1 * g) * distance
Work (friction) = 0.2 * (4kg * 9.8m/s^2) * 2m
Work (friction) = 15.68 Joules
So, the work done by friction is 15.68 Joules.
4) The change in the kinetic energy of the system:
The change in kinetic energy can be calculated using the equation:
Change in kinetic energy = Work (net)
Since the system is moving, the work done on the system is equal to the change in kinetic energy:
Change in kinetic energy = Work (net)
We already calculated the work done by the force F(x) and the work done by friction. Thus, the net work can be calculated by taking the sum of the two:
Net work = Work (F(x)) + Work (friction)
Net work = (8m^2 + 76m) + 15.68
Therefore, the change in kinetic energy of the system is 8m^2 + 76m + 15.68 Joules.