Physics Help >> Index to Physics Homework Help » Slope Problems

Slope problems are usually solved by rotating our frame of reference.

In most problems we assume our "normal" frame of reference: we specify vertical and horizontal directions using "up" or "down", "right" or "left" for example. Consider a toboggan pulled along a virtually frictionless icy surface, by a rope at an angle upward. Since the toboggan is moving horizontally, we would analyze the tension on the rope in terms of its horizontal and vertical components.

• T_v = Tsinø is the vertical component of tension on the rope;
• T_h = Tcosø is the horizontal component of tension on the rope.

In problems involving slope the usual approach is to use a different frame of reference. Since motion is parallel to the slope, directions are specified as parallel to the slope or perpendicular to the slope.

Gravity is resolved into two components:

• F_x = mgsinø
  is the component of gravity parallel to slope: it is that part of weight that pulls an object down the slope.
• F_y = mgcosø
  is the component of gravity perpendicular to the slope: it is usually cancelled by normal force.

Since normal force in most problems is equal in magnitude to mgcosø, then friction = (µN) = µmgcosø. That is, µmgcosø = friction in most slope problems.

Slope Problems

1. A 3.0 m long board has one end raised to a height of 60 cm to form an incline. A 4.0 kg mass is allowed to slide without friction down the entire length of the board.

a. What is the final speed of the mass when it reaches the bottom ?

b. If the mass is replaced with an 8.0 kg mass, what would be the new speed when it reaches the bottom?

2. A wooden block slides directly down an inclined plane, at a constant velocity of 6 m/s.

a. How large is the coefficient of kinetic friction if the plane makes an angle of 25 degrees with the horizontal?

b. If the angle of incline is changed to 10º, how long far will the block slide before coming to a stop?

3. A 10.0 kg box accelerates at 2.00 m/s² as it slides down a ramp that makes an angle of 25.0 degrees with the horizontal. Find the coefficient of friction.

4. A 10.0 kg box is pulled up a 45 degree ramp at a constant velocity by a force of 90.0 N acting parallel to the ramp. Find the coefficient of friction.

5. a 10.0 kg box is pulled up a ramp that is at an angle of 20.0 degrees with the horizontal, with the rope that makes and angle of 30.0 degrees with the ramp. if pulling with a force of 70.0 N causes and acceleration of 2.00 m/s². Find the coefficient of friction.

6. A 10.0 kg box is pulled down a ramp that is at and angle of 45 degrees with the horizontal, with a rope attached to the box that makes a angle of 30.0 degrees with the ramp. if pulling with a force of 10.0 N causes the box to move at a constant velocity what is the coefficient of friction for the box sliding on the ramp?

7. A city planner is working on the redesign of a hilly portion of a city. An important consideration is how steep the roads can be so that even low-powered cars can get up the hills without slowing down. It is given that a particular small car, with a mass of 1100kg, can accelerate on a level road from rest to 21 m/s in 14.0 s. Using this data, calculate the maximum steepness of the hill.

8. An inclined plane that makes an angle of 28° to the horizontal is mounted on wheels. A small block of mass m = 0.9 kg rests on the plane, held there by a coefficient of static friction µ = 0.73. The plane is accelerating to the right. What is the minimum acceleration in order that the block slides down the plane?

9. A 10 kg block is placed on top of an inclined plane 10 m long. The coefficient of static friction is 0.4 and the coefficient of kinetic friction is 0.2. The inclined plane is 30 degrees from the horizontal. a.) What is the acceleration of the block? b.) When will it hit the ground? c.) What is its impact velocity? d.) What is the angle of repose?

10. An object is being pulled up a 15° incline against a frictional coefficient of 0.15, and requires a force of 835 N parallel to the surface of the ramp to move it at a constant speed. What is the weight of the object?

11. The coefficient of static friction between a box and an inclined plane is 0.35.  What is the minimum angle required for the box to begin sliding down the incline?

12. A block is at rest on a sloped surface. Suppose that the angle the slope makes with the horizontal is 38°, and that the frictional force on the block is 27 N. Find the weight of the block.

13. A box of mass (m) is released from rest from the top of a spherical, frictionless surface of radius, R. Find the angle at which the box leaves the surface.

14. A 135 m long ramp is to be built for a ski jump. If a skier starting from rest at the top is to have a speed no faster than 19m/s at the bottom, what should be the maximum angle of inclination?

Answers to Slope Problems

1.

F_x and F_y are components of weight, F_g; F_N is normal force; F_f is friction

Since F_f = 0, and F_N is cancelled by F_y, F_x is the net force

F_net = F_x

ma = mgsinø

a = gsinø [eqn 1]

a.

a = gsinø

ø = sin ^-1( .6 / 3.0) = 11.3º

a =(9.8)sin11.3º = 1.92 m/s²

Applying v_f² = v_i²+ 2ad gives

v_f² = 0 + 2(1.92)(3.0)

v_f = 3.40 m/s

b.

In [eqn 1] we see that the acceleration of the object on the slope (in the absence of friction) does not depend on the object's mass.

For solutions to all the problems on this page click here.