A 20.0-kg crate sits at rest at the bottom of a 20.0 m -long ramp that is inclined at 40 ∘ above the horizontal. A constant horizontal force of 290 N is applied to the crate to push it up the ramp. While the crate is moving, the ramp exerts a constant frictional force on it that has magnitude 65.0 N.
A) What is the total work done on the crate during its motion from the bottom to the top of the ramp?
B) How much time does it take the crate to travel to the top of the ramp?
To find the total work done on the crate during its motion from the bottom to the top of the ramp, we need to calculate the work done by the constant horizontal force and subtract the work done by the frictional force.
A) The work done by a force can be calculated using the formula:
Work = Force × Distance × cos(θ)
Where:
- Force is the magnitude of the force applied or exerted on the object,
- Distance is the distance over which the force is applied, and
- cos(θ) is the cosine of the angle θ between the force and the displacement vectors.
In this case, the force applied is the horizontal force of 290 N, and the distance is the length of the ramp, which is 20.0 m. The angle between the force and displacement vectors is 40°, so θ = 40°.
Using the above formula, we can calculate the work done by the applied force:
Work_applied = Force_applied × Distance × cos(θ)
= 290 N × 20.0 m × cos(40°)
Now, let's calculate the work done by the frictional force. Friction acts opposite to the direction of motion, so the angle between the frictional force and the displacement vectors is 180°, and cos(180°) = -1.
Work_friction = Force_friction × Distance × cos(180°)
= 65.0 N × 20.0 m × cos(180°)
Now, we can calculate the total work done on the crate by subtracting the work done by the frictional force from the work done by the applied force:
Total work = Work_applied + Work_friction
B) To find the time it takes for the crate to travel to the top of the ramp, we can use the concept of average speed and divide the distance traveled by the crate by the average speed.
Average speed = Total distance ÷ Time
The total distance traveled by the crate is the length of the ramp, which is 20.0 m. We can calculate the average speed by dividing this distance by the time taken.
Time = Total distance ÷ Average speed
Therefore, we need to find the average speed of the crate. In this case, the crate is being pushed by a constant horizontal force, so we can use Newton's second law to calculate the acceleration of the crate:
Force_net = mass × acceleration
The net force acting on the crate is the horizontal force minus the frictional force:
Force_net = Force_applied - Force_friction
Using this net force, we can find the acceleration of the crate. Once we have the acceleration, we can calculate the average speed using the kinematic equation:
Average speed = Initial velocity + Final velocity ÷ 2
Since the crate starts from rest, the initial velocity is 0. The final velocity can be calculated using the equation:
Final velocity² = Initial velocity² + 2 × acceleration × distance
Once we have the average speed, we can substitute it into the equation:
Time = Total distance ÷ Average speed
By substituting the values and calculating these equations, we can find the time it takes for the crate to travel to the top of the ramp.
A. F = Fap/Cos A = 290/Cos40 = 379 N. = Force parallel with the ramp.
Work = F*d = 379 * 20
B. F-Fk = M*a.
379 - 65 = 20*a.
a = ?.
d = 0.5a*t^2.
d = 20m, t = ?.