Static Friction along Ramps

The figure shows a box on a ramp. Suppose that the box contains a new flat-screen TV that you’re pushing up the ramp and into your house.

A number of forces are acting on the box, including both gravity and friction, and you need to take both into account. There’s also the force exerted upon the box as you push it up the ramp. So how do you balance all the forces? How much force is needed to get the box moving up the ramp into the house?

To figure that out, you need to calculate the forces along the ramp. There’s the component of the gravitational force along the ramp, which is

But what is the force due to static friction along the ramp?

To find that, you use the equation

What is the normal force, F_n? You already know the answer — it’s

That makes the force due to static friction

The force of static friction points along the ramp, and you’ve calculated the force due to gravity along the ramp. Both of these forces point down the ramp and need to be overcome by the force pushing up the ramp. So in other words, F_push is

Where

is the force due to gravity acting along the ramp, and F_s is the force due to static friction. Plugging in these forces gives you:

Sample question

1. Suppose that the flat-screen TV’s box has a mass of 1.00 x 10² kg, and the ramp has an angle of 23 degrees. What is the force needed to get the box moving up the ramp if the coefficient of static friction is 0.20?

   The correct answer is 563 N.

   1. The equation to use is

      

   2. Plug in the numbers: F_push = 383 + 180 = 563 N.

Practice questions

1. You’re dragging your little brother up the 25-degree wheelchair ramp at the doctor’s office.

   If he has a mass of 40.0 kg and the coefficient of static friction is 0.15, how much force will you need to apply to get him moving?

2. Suppose that you’re struggling to get a 20.0-kg block of ice moving up a 40.0-degree ramp.

   If the coefficient of static friction is a low 0.050, how much force will you need to apply to overcome the weight pulling the block down the ramp and static friction?

Following are answers to the practice questions:

1. 220 N

   1. Calculate the forces you need to overcome: The force due to gravity is

      

      and the force due to friction, F_f, is

      

      You need to find the normal force.

   2. The equation for normal force is

      

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

      

   3. The total force you have to overcome is

      

   4. Plug in the numbers:

      

      The final answer should be rounded to 220 N. (The value for g is only known to two significant figures (9.8) because it varies in the third digit depending on location on Earth.)

2. 130 N

   1. Calculate the forces you need to overcome: The force due to gravity is

      

      and the force due to friction, F_f, is

      

      You need to find the normal force.

   2. The equation for normal force is

      

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

      

   3. The total force you have to overcome is

      

   4. Plug in the numbers:

      

      Note that most of this force is due to the component of the weight along the ramp. The final answer should be rounded to 130 N.