Step-by-step explanation:
(a) To find the work done by the frictional force on the block, we can use the work-energy principle which states that the work done on an object is equal to the change in its kinetic energy. Therefore,
Work done by frictional force = Change in kinetic energy
The initial kinetic energy of the block is given by:
K1 = (1/2)mv1^2 = (1/2)(5 kg)(7 m/s)^2 = 122.5 J
The final kinetic energy of the block is given by:
K2 = (1/2)mv2^2 = (1/2)(5 kg)(4 m/s)^2 = 40 J
Therefore, the change in kinetic energy is:
ΔK = K2 - K1 = 40 J - 122.5 J = -82.5 J
Since the work done by the frictional force is negative, we have:
Work done by frictional force = -|-82.5 J| = 82.5 J
Therefore, the magnitude of the work done by the frictional force on the block is 82.5 J.
(b) To find the magnitude of the average frictional force on the block, we can use the equation:
Average frictional force = Work done by frictional force / Distance traveled
The distance traveled by the block during the 6 seconds is given by:
d = vit + (1/2)at^2
where vi is the initial velocity, t is the time, a is the acceleration, and d is the distance.
Since the surface is smooth before the rough patch, the block moves with constant velocity and the acceleration is zero. Therefore,
d = vit = (7 m/s)(6 s) = 42 m
After the rough patch, the block slows down from 7 m/s to 4 m/s in 6 seconds. Therefore, the acceleration of the block is given by:
a = (v2 - v1) / t = (4 m/s - 7 m/s) / 6 s = -0.5 m/s^2
Using this acceleration, we can find the distance traveled by the block after the rough patch:
d' = vit + (1/2)at^2 = (4 m/s)(6 s) + (1/2)(-0.5 m/s^2)(6 s)^2 = 12 m
Therefore, the total distance traveled by the block is:
d_total = d + d' = 42 m + 12 m = 54 m
Now, we can find the average frictional force on the block:
Average frictional force = Work done by frictional force / Distance traveled
= 82.5 J / 54 m
≈ 1.53 N
Therefore, the magnitude of the average frictional force on the block is approximately 1.53 N.