Interactive Lecture Demonstration 7 – Truck Parked on a Hill

In this demonstration, a block ("truck") is "parked" (i.e. is in equilibrium and is not moving) on a hill. A real truck would be held from sliding down the hill by the force of static friction of the road surface on the wheels. Our truck, though, is stopped from sliding by a barrier placed downhill of the truck. The angle of the hill is 37 degrees above horizontal. The mass of the truck is 435 grams.

The purpose of this demonstration is to determine the magnitudes of all the forces acting on the truck, and then to test our predictions. This problem will give us practice in doing net force problems when forces act in two dimensions.

1. As usual, start with a diagram of the situation. Next to your diagram, draw the force diagram for the truck.

2. Now we need to choose positive directions. Since the truck is not accelerating, we may choose any two mutually perpendicular directions. We should choose the directions that make solving the problem easiest. This means choosing directions so that most of the forces in the problem point along those directions, so we will need to break the fewest number of forces into components. Indicate next to your force diagram your choices of positive directions.

3. We hope you chose positive directions that point parallel to the slope and perpendicular to it. Now we must determine the components of the force that does not lie along either of those directions – the weight of the truck. Draw a right triangle for that force. The total weight force must be the hypotenuse of the triangle. The other two sides of the triangle must be along the two directions you chose already – parallel and perpendicular to the slope. You may find it helpful to draw the total weight vector first, then draw in the other two sides by matching their directions with your diagram of the hill. Remember to include directional arrows on all three sides of your vector diagram. Then label in your triangle the angle that is the same as the slope's angle (37 degrees). Finally, label all three sides of the triangle with their magnitudes, expressed in terms of the magnitude of the hypotenuse (i.e. mg) and appropriate trigonometric functions of the angle of the slope.  

4. Now write the two net force equations for this problem (one for the parallel direction, one for perpendicular). This should be easy – just read the forces from your force diagram. Be sure to include in each equation the component of every force that lies along the direction for your equation. Put a + sign in front of components that point in the + direction, a – sign in front of components that point in the – direction.

5. The rest of the problem is algebra. We want to solve for the magnitudes of the two normal forces acting on the truck (one from the hill, one from the barrier downhill of the truck). Do that now. After you've solved for the unknowns using variables, plug in numbers with units and get your answers.

6. Now we will test our results. We will apply a tension force to pull the truck parallel to the ramp, using strings and hanging weights. How much mass should we hang to create a tension force that's equal in magnitude to the normal force currently being produced by the barrier ? What will happen to the truck when we hang the weights ? What should we be able to do with the barrier after we apply the tension force ?

7. Now we'll test our calculation for the normal force of the hill on the truck. We will apply another tension force, from a string that pulls perpendicular to the slope. How much mass should we hang from the other end of the string to create a tension force equal in magnitude to the force currently being produced by the hill ? What will happen to the truck when we hang the weights ? What should we be able to do with the hill after we apply the tension force ?