A 10.0 kg crate is pulled up a rough incline with an initial speed of 1.5 m/s. The pulling force is 100.0N parallel to the incline, which makes an angle of 15.0º with the horizontal. Assuming the coefficient of kinetic friction is 0.40 and the crate is pulled a distance of 7.5 m, find the following:
a. the work done by earth's gravity on the crate
b. the work done by the force of friction on the crate
c. the work done by the puller on the crate
d. the change in kinetic energy of the crate
e. the speed of the crate after it is pulled 7.5 m.
To solve this problem, we need to break it down into several steps. Let's start by finding the work done by different forces on the crate.
a. The work done by earth's gravity on the crate:
The work done by gravity can be calculated using the formula:
Work = force × distance × cos(θ)
Where:
force = weight = mass × gravity (where gravity is approximately 9.8 m/s^2)
distance = 7.5 m
θ = angle between the force and the displacement (in this case, it is the angle of the incline with the horizontal)
So, the work done by gravity is:
Work_gravity = (mass × gravity) × distance × cos(θ)
Plug in the values:
Work_gravity = (10.0 kg × 9.8 m/s^2) × 7.5 m × cos(θ)
b. The work done by the force of friction on the crate:
The work done by friction is given by the equation:
Work_friction = force_friction × distance
The force of friction is calculated using the formula:
force_friction = coefficient of friction × normal force
The normal force can be calculated as:
normal force = mass × gravity × cos(θ)
So, the force of friction is:
force_friction = coefficient of friction × (mass × gravity × cos(θ))
Therefore, the work done by friction is:
Work_friction = (coefficient of friction × (mass × gravity × cos(θ))) × distance
c. The work done by the puller on the crate:
The work done by the puller can be calculated using the formula:
Work_puller = force_puller × distance × cos(θ)
Given that the pulling force is 100.0 N and the angle θ is 15.0º, we can calculate:
Work_puller = 100.0 N × distance × cos(θ)
d. The change in kinetic energy of the crate:
The change in kinetic energy is equal to the work done by the net force acting on the object. In this case, the net force is the force of the puller minus the force of friction:
Net_force = force_puller - force_friction
Now we can calculate:
Work_net = Net_force × distance
Change_in_kinetic_energy = Work_net
e. The speed of the crate after it is pulled 7.5 m:
We can find the final speed of the crate using the equation:
Final_speed^2 = Initial_speed^2 + 2 × acceleration × distance
In this case, the acceleration can be calculated using the net force:
acceleration = Net_force / mass
Once we have the acceleration, we can substitute the values to find the final speed.
I hope this explanation helps you in solving the problem step by step.