A 5.30 kg package slides 1.57 m down a long ramp that is inclined at 11.8^\circ below the horizontal. The coefficient of kinetic friction between the package and the ramp is mu_k = 0.320
Calculate the work done on the package by the normal force.
If the package has a speed of 2.17 m/s at the top of the ramp, what is its speed after sliding the distance 1.57 m down the ramp?
Why the name changes?
To calculate the work done on the package by the normal force, we first need to find the magnitude of the normal force.
The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. It acts in the opposite direction of the gravitational force on the object.
To find the magnitude of the normal force, we can use the following equation:
Normal force = Weight of the object - Component of weight parallel to the ramp
Weight of the object = mass × gravitational acceleration
Weight of the object = 5.30 kg × 9.8 m/s^2
Next, we need to find the component of weight parallel to the ramp. This component is determined by the angle of inclination, 11.8 degrees in this case.
Component of weight parallel to the ramp = Weight of the object × sin(angle of inclination)
Now, we can find the normal force:
Normal force = Weight of the object - Component of weight parallel to the ramp
With the normal force determined, we can now calculate the work done on the package by the normal force.
Work done = Force × Distance × cos(angle between force and displacement)
Since the normal force and displacement are in the same direction (perpendicular to the ramp), the angle between them is 0 degrees. In this case, cos(angle between force and displacement) = 1.
Next, we can calculate the speed of the package after sliding the distance down the ramp.
The initial speed of the package at the top of the ramp, 2.17 m/s, can be considered as the final speed after sliding 1.57 m down the ramp.
To calculate the final speed, we need to consider the conservation of mechanical energy. The work done by friction is equal to the change in mechanical energy.
Work done by friction = Change in mechanical energy
The work done by friction can be calculated using the equation:
Work done by friction = Force of friction × Distance
Force of friction = coefficient of kinetic friction × Normal force
Using the work-energy principle, the change in mechanical energy can be expressed as:
Change in mechanical energy = Work done by gravity + Work done by friction
Since the package is sliding down the ramp, the work done by gravity is negative due to the opposite direction of displacement.
Change in mechanical energy = -Work done by gravity + Work done by friction
Finally, we can find the final speed of the package by using the equation:
Change in mechanical energy = (1/2) × mass × (final speed)^2 - (1/2) × mass × (initial speed)^2
Solving this equation for the final speed gives us the desired answer.