Part 1
Since there is no friction, there is n force opposing the downward movement of the block. The freebody diagram of the block is shown below
The force pulling the block down the ramp = mgSinθ
From the information given,
θ = 30
m = 9
g = 9.8
Thus,
Force = 9 x 9.8Sin30 = 44.1
Recall, Force = mass x acceleration
acceleration = force/mass = 44.1/9 = 4.9 m/s^2
The block accelerated to the bottom at 4.9 m/s^2
The block started from rest, thus, initial velocity = 0
Distance travelled = 11 m
We want to calculate the final velocity at the bottom. We would apply the formula,
v^2 = u^2 + 2as
where
v = final velocity
u = initial velocity
a = acceleration
s = diatance
Thus,
v^2 = 0^2 + 2 x 4.9 x 11 = 107.8
v = square root of 107.8 = 10.38
The speed of the block at the bottom of the ramp = 10.38 m/s
b) The movement of the block on the floor is opposed by friction. Fr represents frictional force. Fa is the frorce moving the block forward.
Recall, Fa = 44.1N
Frictional force = normal reaction x coefficient of friction
The block travels 20.8 m on the floor before coming to rest. Thus,
initial velocity = 10.38
final velocity = 0
We would find the acceleration by applying the formula
v^2 = u^2 - 2as
The negative sign is because the block was decelerating. Thus,
0^2 = 10.38^2 - 2 x a x 20.8
10.38^2 = 2 x a x 20.8
107.744 = 41.6
a = 107.744/41.6 = 2.59
Force with which the block moved on the floor = 9 x 2.59 = 23.31N
Recall, frictional force = 9 x 9.8 x coefficient of friction
Thus,
44.1 - 88.2 x coefficient of friction = 23.31
88.2 x coefficient of friction = 44.1 - 23.31 = 20.79
coefficient of friction = 20.79/88.2
coefficient of friction = 0.24
c) The height of the block above the ground is calculated by
height = 11 cos30 = 9.53
Potential energy = mgh
Potential energy of block at the top = 9 x 9.8 x 9.53 = 840.55 J
Kinetic energy of the block when it came to a stop = 0
Thus, Mechanical energy lost due to friction = 840.55 - 0 = 840.55 J