Acceleration along Ramps Including Friction

By Steven Holzner

Suppose that you have a crate that has been mistakenly placed near the top of a long ramp, and it starts sliding down that ramp. How about calculating its acceleration down the incline?

The object is sliding down the ramp — you’re not pushing it — which means the force of kinetic friction is opposing (not adding to) the component of gravity along the ramp. So the net force on the crate is

image1.png

Because a = F/m, the acceleration of the crate is

image2.png

Notice that the crate’s mass does not affect its acceleration because mass cancels out of the final equation.

Sample question

  1. A plastic crate slips down a 19-degree ramp with a coefficient of kinetic friction of 0.10. What is its acceleration as it slides?

    The correct answer is 2.3 m/s2.

    1. You can use the equation

      image4.png

    2. Plug in the numbers: a = 3.2 – 0.93 = 2.3 m/s2.

Practice questions

  1. You’re dragging a suitcase up a ramp into a luxury hotel when it gets away from you. If the angle of the ramp is 31 degrees and the coefficient of kinetic friction is 0.1, what is the suitcase’s acceleration down the ramp?

  2. You drop a 5.0-kg box on a ramp of 12 degrees, and the coefficient of kinetic friction is 0.15. Will the box slide down the ramp?

  3. You drop a 1.0-kg book on a 15-degree ramp, and the coefficient of kinetic friction is 0.30. Will the book slide down the ramp?

  4. A refrigerator breaks away from the movers and slides down a 23-degree ramp that has a coefficient of kinetic friction of 0.25. What is its acceleration?

Following are answers to the practice questions:

  1. 4 m/s2

    1. Calculate the forces on the suitcase: The force due to gravity is

      image5.png

      and the force due to kinetic friction is

      image6.png

      You need to find the normal force.

    2. The equation for the normal force is

      image7.png

      Use the normal force to calculate the force due to kinetic friction:

      image8.png

    3. The net force on the suitcase along the ramp is

      image9.png

    4. Divide the net force along the ramp by m to get the acceleration:

      image10.png

    5. Plug in the numbers:

      image11.png

  2. Yes, the box will slide.

    1. Calculate the forces on the box, and if the force down the ramp is larger than the force up the ramp, the box will slide. The force down the ramp is

      image12.png

    2. The force due to kinetic friction pointing up the ramp is

      image13.png

      which means you need to find the normal force.

    3. The equation for the normal force is

      image14.png

      Use the normal force to calculate the force due to kinetic friction:

      image15.png

    4. The net force on the suitcase down the ramp is

      image16.png

    5. Plug in the numbers

      image17.png

      down the ramp, so the box will slide.

  3. No, the book will not slide.

    1. Calculate the forces on the book, and if the force down the ramp is larger than the force up the ramp, the book will slide. The force down the ramp is

      image18.png

    2. The force due to kinetic friction pointing up the ramp is

      image19.png

      which means you need to find the normal force.

    3. The equation for the normal force is

      image20.png

      Use the normal force to calculate the force due to kinetic friction:

      image21.png

    4. The net force on the book down the ramp is

      image22.png

    5. Plug in the numbers

      image23.png

      along the ramp, so the book will not slide.

  4. 1.5 m/s2 down the ramp

    1. Calculate the forces on the refrigerator: The force down the ramp is

      image24.png

      and the force due to kinetic friction pointing up the ramp is

      image25.png

      You need to find the normal force.

    2. The equation for the normal force is

      image26.png

      Use the normal force to calculate the force due to kinetic friction:

      image27.png

    3. The net force on the refrigerator down the ramp is

      image28.png

    4. Divide by m to get the acceleration:

      image29.png

    5. Plug in the numbers:

      image30.png