A student decides to move a box of books into her dormitory room by pulling on a rope attached to the box. She pulls with a force of 100.0 N at an angle of 25.0° above the horizontal. The box has a mass of 25.0 kg, and the coefficient of kinetic friction between box and floor is 0.300.
(a) Find the acceleration of the box.
(b) The student now pulls the moving box up a 10.0° incline, keeping her 100.0 N force directed at 25.0° above the line of the incline. If the coefficient of friction is unchanged, what is the new acceleration of the box?
break up the rope force into two components. The vertical component reduces the normal force (mg-verticalforce), so friction is reduced.
Then use the horizontal component of force to set equal to ma + frictionforce.
To find the acceleration of the box in both situations, we need to consider the forces acting on it.
(a) In the first scenario, the box is being pulled horizontally. The forces acting on the box are the applied force (pulling force) and the kinetic friction force. The gravitational force can be ignored since it acts vertically and does not affect the horizontal motion.
The formula to calculate the net force acting on the box is:
Net force = Applied force - Frictional force
The applied force can be resolved into horizontal and vertical components as follows:
Applied force (horizontal) = Force * cos(angle)
Applied force (vertical) = Force * sin(angle)
The frictional force is given by:
Frictional force = Coefficient of kinetic friction * Normal force
The normal force is the force exerted by the surface on the box and is equal to the weight of the box since the box is on a horizontal surface:
Normal force = Weight = mass * gravitational acceleration
Now, we can calculate the individual forces and plug them into the formula to find the net force and then the acceleration.
(b) In the second scenario, the box is being pulled up an incline. The forces acting on the box are the applied force, the gravitational force, and the frictional force. We need to resolve the applied force into components again, this time along and perpendicular to the incline.
The formula to calculate the net force acting on the box is:
Net force = Applied force (along the incline) - Weight (parallel to the incline) - Frictional force
The applied force along the incline can be calculated as:
Applied force (along the incline) = Force * sin(angle)
The weight parallel to the incline can be calculated as:
Weight (parallel to the incline) = Weight * sin(incline angle)
Again, the frictional force is given by:
Frictional force = Coefficient of kinetic friction * Normal force
The normal force in this case is equal to the component of the weight perpendicular to the incline, which can be calculated as:
Normal force = Weight * cos(incline angle)
Now, we can calculate the individual forces and plug them into the formula to find the net force and then the new acceleration.