The Inclined Plane
1. Explain the formulas for two components of the gravitational force on an object that is placed on an inclined plane:
Fperpendicular=m⋅g⋅cos(θ) Fparallel=m⋅g⋅sin(θ)
2. A toy elephant with a mass of 2 kg rolls down a frictionless plane. The plane has an inclination of 30°.
a) Sketch the situation (keep it simple).
b) Calculate the gravitational force acting on the cart. Include an arrow signifying the force in your sketch.
c) Calculate the component of the gravitational force parallel to the plane. Include an arrow signifying the force in your sketch.
d) Calculate the component of the gravitational force perpendicular to
the plane. Include an arrow signifying the force in your sketch.
3. Answer the questions in exercise 2 for a toy elephant with a mass of 2 kg, in the case that the plane has an inclination of 5°.
4. A mass, m1, of 3 kg is placed on an inclined plane and connected to another mass, m2, of 2 kg via a string. The plane has an inclination of 30°.
a) Calculate the gravitational force on m2. Include an arrow signifying the force on the illustration above.
b) With what force does m2 affect m1? With what force does m1 affect m2? Include these forces on the illustration above.
c) Calculate the gravitational force on m1.
d) Calculate the component of the gravitational force on m1 parallel to the plane. Include this force on the illustration above.
e) Use the answers from b and d to find the total force on the m1 parallel to the plane.
f) Does m1 slide up or down the plane?
g) Calculate the acceleration of m1 based on the above.
h) Calculate the change in distance covered by m1at t = 1 s, if the box starts at rest at t = 0.
5. Answer the questions in exercise 4, in the case that the plane has an inclination of 30°, m1 is 3 kg and m2 is 1 kg.
6. Take a look at the situation sketched below. Evaluate the forces involved parallel to the planes and find out how the system will behave, if the carts start out at rest in the situation shown.
7. How will the system in 6 behave, if both carts have the same mass?
8. A mass of 3 kg slides down a frictionless inclined plane that has a height, h, of 2 m. It starts out at rest at the top.
a) Calculate the potential energy at the top of the plane relative to the ground.
b) Calculate the kinetic energy that a mass has, when the mass reaches them bottom of the frictionless plane.
c) Calculate the velocity corresponding to the kinetic energy found in b.
9. Answer the questions in exercise 8, in the case that the plane has a height of 1 m and the mass of the object sliding down the plane is 2 kg.
10. We now look at a setup that resembles the setup from exercise 4 but with slightly different values. In this case, m1 will slide down the plane. The height of the plane is 1.5 m. When m1(5 kg) is at the top of the plane m2 (1 kg) stands on the ground, and when m2 has reached the top, m1 has travelled halfway down the plane – so m1 is 0.75 m above the ground.
a) Calculate the potential and kinetic energy, when m2 is still on the ground, and m1 is at rest at the top of the plane.
b) Calculate the potential and kinetic energy of the two masses, when the m2 has reached the top, and m1 is rushing down the plane.
