In a circus performance, a monkey on a sled is given an initial speed of 3.2 m/s up a 31° incline. The combined mass of the monkey and the sled is 20.7 kg, and the coefficient of kinetic friction between the sled and the incline is 0.20. How far up the incline does the sled move?
KE= PE+W(fr)
KE=m•v²/2
PE= m•g•h=m•g•s•sinα
W(fr)=F(fr) •s= μ•m•g•cosα•s
m•v²/2= m•g•s•sinα+ μ•m•g•cosα•s.
s= v²/2g(sinα+ μ•cosα)
To find out how far up the incline the sled moves, we can use the concept of work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
First, let's calculate the work done by the net force acting on the sled. The net force can be found by subtracting the force of friction from the component of the gravitational force parallel to the incline.
1. Calculate the gravitational force parallel to the incline:
The gravitational force parallel to the incline can be found using the formula:
F_parallel = m * g * sin(theta), where
m = mass of the monkey and the sled = 20.7 kg
g = acceleration due to gravity = 9.8 m/s^2
theta = angle of the incline = 31°
F_parallel = 20.7 kg * 9.8 m/s^2 * sin(31°)
F_parallel ≈ 101.8 N
2. Calculate the force of friction:
The force of friction can be found using the formula:
F_friction = coefficient of kinetic friction * normal force, where
coefficient of kinetic friction = 0.20
normal force = m * g * cos(theta)
normal force = 20.7 kg * 9.8 m/s^2 * cos(31°)
normal force ≈ 176.8 N
F_friction = 0.20 * 176.8 N
F_friction ≈ 35.4 N
3. Calculate the net force:
The net force can be found by subtracting the force of friction from the gravitational force parallel to the incline:
Net force = F_parallel - F_friction
Net force = 101.8 N - 35.4 N
Net force ≈ 66.4 N
4. Calculate the work done:
The work done is equal to the force applied to the sled multiplied by the distance over which the force is applied:
Work = force * distance
Since the force and distance are acting in the same direction, the work done is positive.
We can rearrange the work formula to find the distance:
Distance = Work / Force
Distance = Work / Net force
To find the work, we need to calculate the change in kinetic energy. The sled starts with an initial speed and will come to a stop at some distance up the incline.
5. Calculate the change in kinetic energy:
The change in kinetic energy can be found using the formula:
ΔKE = (1/2) * m * (vf^2 - vi^2), where
m = mass of the monkey and the sled = 20.7 kg
vf = final velocity = 0 m/s (since the sled comes to a stop)
vi = initial velocity = 3.2 m/s
ΔKE = (1/2) * 20.7 kg * (0^2 - 3.2^2)
ΔKE ≈ -105.2 J
Note: The negative sign indicates that the kinetic energy decreases.
6. Calculate the work done:
The work done is equal to the change in kinetic energy:
Work = ΔKE
Work = -105.2 J
7. Calculate the distance:
Finally, we can calculate the distance using the work formula:
Distance = Work / Net force
Distance = -105.2 J / 66.4 N
Distance ≈ -1.583 m
The negative distance indicates that the sled moves in the opposite direction of the incline. To get the actual distance, we take the absolute value:
Distance = | -1.583 m |
Therefore, the sled moves approximately 1.583 meters up the incline.