A mass of 8 kg with initial velocity 18 m/s
travels through a wind tunnel that exerts a
constant force 5 N for a distance 0.5 m. It
climbs a frictional incline of height 2
clined at an angle 16
◦
, then moves along a
second frictional surface of coefficient 0.19 before coming to rest.
The acceleration of gravity is 9.8 m/s
2
.
µ
µ
m
L h
2
F
S
x
1
θ
vο
If the first frictional surface has a coefficient
of 0.24 for a distance 0.8 m, how far does it
slide along the second frictional region before
coming to rest?
Answer in units of m
To find the distance that the mass slides along the second frictional surface before coming to rest, we need to determine the force of friction acting on the mass and then use it to calculate the distance traveled.
First, let's calculate the force of friction acting on the mass on the first frictional surface using the given coefficient of friction (µ = 0.24):
Force of friction (F_friction1) = coefficient of friction (µ) * normal force (N)
The normal force (N) can be calculated using the mass of the object (m) and the acceleration due to gravity (g):
Normal force (N) = mass (m) * acceleration due to gravity (g)
Normal force (N) = 8 kg * 9.8 m/s^2 = 78.4 N
Now, we can calculate the force of friction:
F_friction1 = 0.24 * 78.4 N = 18.82 N
Next, let's calculate the work done by the force of friction on the first frictional surface:
Work (W) = force (F) * distance (d)
Work (W) = F_friction1 * 0.8 m = 18.82 N * 0.8 m = 15.056 J
Since work is directly related to the change in kinetic energy, we can use the work-energy principle to determine the change in the object's kinetic energy on the first frictional surface:
Change in kinetic energy (ΔKE) = Work (W)
ΔKE = 15.056 J
The change in kinetic energy can be calculated using the following formula:
ΔKE = (1/2) * mass (m) * (final velocity^2 - initial velocity^2)
ΔKE = (1/2) * 8 kg * (0 - 18 m/s)^2
0.5 * 8 kg * (18 m/s)^2 = 324 J
Since the mass comes to rest on the second frictional surface, the change in kinetic energy (ΔKE) is equal to zero.
Therefore, we can equate the work done on the second frictional surface to the negative of the change in kinetic energy:
Work (W) = -ΔKE
Work (W) = 0
Now, let's determine the force of friction on the second frictional surface using the given coefficient of friction (µ = 0.19):
Force of friction (F_friction2) = coefficient of friction (µ) * normal force (N)
Since the mass is at rest, the normal force on the second frictional surface is equal to the weight of the object:
Normal force (N) = mass (m) * acceleration due to gravity (g)
Normal force (N) = 8 kg * 9.8 m/s^2 = 78.4 N
F_friction2 = 0.19 * 78.4 N = 14.896 N
Finally, we can determine the distance (d) traveled on the second frictional surface before coming to rest:
Work (W) = force (F) * distance (d)
Distance (d) = Work (W) / force (F_friction2)
Distance (d) = 0 / 14.896 N = 0 m
Therefore, the mass does not slide along the second frictional region before coming to rest. The distance traveled on the second frictional surface is 0 m.