A 40N crate starting at rest slides down a rough 6.0m long ramp inclined at 30 degrees with the horizontal. The force of friction between the crate and ramp is 6.0N. Using the work-kinetic energy theorem, find the velocity of the crate at the bottom of the incline.
To find the velocity of the crate at the bottom of the incline using the work-kinetic energy theorem, we need to calculate the net work done on the crate.
Step 1: Determine the work done by the force of gravity.
The work done by the force of gravity is given by W_gravity = mgh, where m is the mass of the crate, g is the acceleration due to gravity, and h is the height of the incline. In this case, the height of the incline is h = 6.0m * sin(30°) = 3.0m.
The mass of the crate can be calculated using the formula F = ma, where F is the force and a is the acceleration. The force F is equal to the weight of the crate, which is equal to mg, where g is the acceleration due to gravity (9.8m/s^2).
So, the mass of the crate is m = F/g = 40N / 9.8m/s^2 = 4.08 kg.
Therefore, the work done by the force of gravity is W_gravity = (4.08kg)(9.8m/s^2)(3.0m) = 119.9J.
Step 2: Determine the work done by the force of friction.
The work done by the force of friction is given by W_friction = F_friction * d * cos(θ), where F_friction is the force of friction, d is the distance traveled, and θ is the angle between the direction of the force and the displacement. In this case, the angle θ is 180° since the force of friction opposes the motion of the crate.
Therefore, the work done by the force of friction is W_friction = (6.0N)(6.0m)(cos(180°)) = -36J.
Note that the work done by friction is negative because it acts in the opposite direction of the displacement.
Step 3: Calculate the net work done.
The net work done is given by the sum of the individual works done: W_net = W_gravity + W_friction = 119.9J + (-36J) = 83.9J.
Step 4: Use the work-kinetic energy theorem to find the final velocity.
According to the work-kinetic energy theorem, the net work done is equal to the change in kinetic energy of the object: W_net = ΔKE = KE_final - KE_initial.
Since the crate starts at rest, the initial kinetic energy is zero: KE_initial = 0.
Therefore, the equation becomes: 83.9J = KE_final - 0.
Simplifying, we find that the final kinetic energy is equal to the net work done: KE_final = 83.9J.
Finally, we can find the velocity using the equation KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Rearranging the equation, we have: v = √(2KE / m) = √(2 * 83.9J / 4.08kg) ≈ 5.38 m/s.
Therefore, the velocity of the crate at the bottom of the incline is approximately 5.38 m/s.
To find the velocity of the crate at the bottom of the incline, we can use the work-kinetic energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
The work done on the crate is equal to the force applied to it (which is the component of its weight parallel to the incline) multiplied by the displacement along the incline. In this case, the force applied parallel to the incline is the net force acting on the crate, which is the difference between the force of gravity parallel to the incline and the force of friction:
Net force = force of gravity - force of friction
The force of gravity parallel to the incline can be calculated by multiplying the weight of the crate by the sine of the angle of the incline:
Force of gravity parallel to incline = weight of crate * sin(angle of incline)
The net force can then be calculated:
Net force = (weight of crate * sin(angle of incline)) - force of friction
To find the work done on the crate, we can multiply the net force by the displacement along the incline:
Work done = net force * displacement
Now, according to the work-energy theorem, this work done on the crate is equal to the change in its kinetic energy. Since the crate starts from rest, its initial kinetic energy is zero. Therefore:
Work done = change in kinetic energy
Finally, we can equate the work done to the change in kinetic energy and solve for the final velocity:
Work done = (1/2) * mass * velocity^2
By substituting the values we know into the equation, we can find the velocity of the crate at the bottom of the incline.