Help with a question form 1984 physics B AP exam

[Petrikovski]

Two 10-kilogram boxes are connected by a massless string that passes over a massless frictionless pulley. The bxes remain at rest, with one on the right hanging vertically and the one n the left 2.0 meters from the bottom of an inclined plane that makes an angle of 60 degrees with the horizontal. The coefficients of kinetic and static friction between the box on the incline and the plane are .15 and .30 respectively. You may use g = 10 m/s^2, sin60 = .87 and cos60 = .5

The string is then cut and the left-hand box slides down the inclined plane

d) Determine the amount of mechanical energy that is converted into thermal energy during the slide to the bottom?

e) Determine the kinetic energy of the left-hand box when it reaches the bottom of the plane. I assume for this i use KE = 1/2mv^2 but how do i get V?

 

[Astronuc]

Well the box is sliding down the incline under influence of gravity, however there is a resistive force of friction. One calculates the net force.

From force, one obtains the acceleration.

From acceleration one can apply the appropriate relationship between velocity, distance and acceleration. The initial velocity is zero.

The thermal energy is related force of friction applied over distance.

 

[quicknote]

To determine the amount of mechanical energy converted into thermal energy during the slide, we can use the principle of conservation of energy. The initial mechanical energy of the system is equal to the final mechanical energy plus the energy lost due to friction. The initial mechanical energy is the potential energy of the box on the inclined plane, which can be calculated using the formula PE = mgh, where m is the mass of the box, g is the acceleration due to gravity, and h is the height of the inclined plane (2.0 meters). This gives us an initial potential energy of 200 joules (J).

The final mechanical energy is the kinetic energy of the box when it reaches the bottom of the plane. To calculate this, we can use the formula KE = 1/2mv^2, where m is the mass of the box and v is the velocity of the box at the bottom of the plane. To find the velocity, we can use the equation of motion v^2 = u^2 + 2as, where u is the initial velocity (in this case, 0 m/s), a is the acceleration (which is due to gravity and can be calculated using the angle of the inclined plane), and s is the distance traveled (which is 2.0 meters). This gives us a final velocity of approximately 5.5 m/s. Plugging this into the formula for kinetic energy, we get a final kinetic energy of approximately 137.5 J.

Therefore, the amount of mechanical energy converted into thermal energy during the slide is 200 J - 137.5 J = 62.5 J.

To determine the kinetic energy of the left-hand box when it reaches the bottom of the plane, we can use the same formula as above: KE = 1/2mv^2. We have already calculated the mass of the box and the velocity at the bottom of the plane, so we can simply plug these values in to get the kinetic energy. This gives us a kinetic energy of approximately 137.5 J, as calculated above.