Part E Use the general expression for conservation of energy

Part E

Use the general expression for conservation of energy

Ui+Ki+Wother=Uf+Kf

to write an equation relating the initial and final kinetic and potential energies and the work done by friction.

Enter your equation symbolically using the variables given in the problem introduction (m,g,,fk,v and d). Do not use the variable h. Enter the left side of the equation and the right side of the equation separated by a comma. Do not use an equal sign.

Solution

Here, you need to note the forces which would be doing some work on the toolbox as it slides down the roof. Then we can form the equation using the principle of conservation of energy.

Plus, you would also need to keep the following into consideration:

a.) The toolbox will have two forces acting on it while it slides: The gravitational force trying to pull the box down and the frictional force trying to oppose the motion.

b.) At the top of the roof, the box will have no Kinetic Energy as it started from rest, but will have some potential energy with respect to the lower edge of the roof.

c.) At the lower edge of the roof, the toolbox will have only Kinetic energy and no potential energy [We are using the lower edge of the roof as the reference level]

Now we will use the above to solve the given problems:

For velocity: The net acceleration of the block down the roof would be resultant of gravitational acceleration and the deceleration offered by friction. Hence we have:

acceleration = gSin - Fk/m =5.7662 - 2.588 = 3.1782 m/s^2

Therefore the velocity over the distance of d, can be given as:

V^2 = u^2 + 2aS = 0 + 2(gSin - Fk/m)d = 27.0147

That is V = Sqrt[2(gSin - Fk/m)d] = 5.1976 m/s

For the equation: Now for the equation, we have:

Uf = 0; Kf = 0.5mv^2

Ui = mgdSin; Ki = 0

Also Work done by friction = Fkd

Therefore, for the form Ui + Ki + Wother = Uf + Kf, we get:

mgdSin + 0 - Fkd = 0 + m(gSin - Fk/m)d [Since the final PE is zero and initial KE is zero, also, the motion of the block happens against the direction of friction, hence the work done by friction is negative]