A 475 gram box is given a push and it then slides up and back down a ramp with a 35.0° incline. The coefficient of friction is 0.30. (a) Determine the rate of deceleration as the box slides up the ramp. (b) Determine the rate of acceleration as the box slides back down the ramp. (c) Determine the amount of applied force necessary to push the box up the ramp at a steady speed.
To solve this problem, we need to use Newton's Second Law of Motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F=ma). We'll break the problem down into three parts: determining the net force on the box while sliding up the ramp, while sliding back down the ramp, and while pushing the box up the ramp at a steady speed.
a) Determining the rate of deceleration as the box slides up the ramp:
The net force acting on the box while sliding up the ramp is the force of gravity (weight) minus the force of friction. We can express this using the equation:
Net force = Weight - Friction force
Weight = mass * gravitational acceleration (g)
Friction force = coefficient of friction * normal force
The normal force is the force exerted by the ramp perpendicular to the surface. It can be calculated using the equation:
Normal force = Weight * cos(angle)
Given:
Mass of the box (m) = 475 grams = 0.475 kg
Gravitational acceleration (g) = 9.8 m/s²
Angle of the incline (theta) = 35.0°
Coefficien of friction (mu) = 0.30
1. Calculate the normal force:
Normal force = Weight * cos(theta)
Weight = m * g
Normal force = m * g * cos(theta)
2. Calculate the friction force:
Friction force = mu * Normal force
3. Calculate the net force acting on the box:
Net force = Weight - Friction force
4. Calculate the deceleration:
Net force = m * a
a (deceleration) = Net force / m
b) Determining the rate of acceleration as the box slides back down the ramp:
The direction of motion is now opposite to the force of friction, so we need to take that into account. The net force is the force of gravity (weight) plus the force of friction this time.
1. Calculate the normal force (same as in part a).
2. Calculate the friction force:
Friction force = mu * Normal force
3. Calculate the net force acting on the box:
Net force = Weight + Friction force
4. Calculate the acceleration (as the box is moving downward, acceleration will be positive):
Net force = m * a
a (acceleration) = Net force / m
c) Determining the amount of applied force necessary to push the box up the ramp at a steady speed:
When the box moves up the ramp at a steady speed, the net force is zero, which means the applied force balances the force of friction.
1. Calculate the normal force (same as in part a).
2. Calculate the friction force:
Friction force = mu * Normal force
3. Calculate the applied force:
Applied force = Friction force
These calculations will allow you to determine the rate of deceleration, rate of acceleration, and the applied force necessary to push the box up the ramp at a steady speed. Substitute the given values into the equations to get the numerical answers.