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Calculating How Far an Object Will Slide Down an Inclined Surface

You can use physics to calculate how far an object will slide down an inclined surface, such as a ramp. For example, say you and your friends are pushing a refrigerator up a ramp onto a moving van, when suddenly your combined strength gives out and the refrigerator begins to plummet back down the 3.0-meter ramp. As you watch with dismay, it picks up speed. A car is parked behind the ramp, only 7.2 meters away. Will the errant refrigerator smash it?

Assuming that the ramp and the ground both have the same kinetic coefficient of friction and that the refrigerator starts to slide from the top of the ramp, you need to calculate how far the refrigerator will slide. You quickly pull out a pencil and paper and draw a figure showing the refrigerator as it slides down the 3.0-meter ramp.

All the forces acting on an object sliding down a ramp.
All the forces acting on an object sliding down a ramp.

When an object slides downward, the forces acting on it are different from the forces used to push it up the ramp (see the figure). With the fridge, there’s no Fpush force to push it up the ramp. Instead, the component of the refrigerator’s weight acting along the ramp pulls the refrigerator downward. And while the fridge slides down, friction opposes that force. You also need to know the coefficient of kinetic friction,

image1.png

For this example, the coefficient of kinetic friction is 0.15.

So what force accelerates the refrigerator downward? The weight acting along the ramp is

image2.png

and the normal force is

image3.png

which means that the force of kinetic friction is

image4.png

The net force accelerating the refrigerator down the ramp, Facceleration, is the difference between the component of the refrigerator’s weight along the ramp and the frictional force opposing it:

image5.png

Note that you subtract FF, the force due to friction, because that force always acts to oppose the force causing the object to move. Plugging in the numbers gives you

image6.png

The net force pulling the refrigerator down the ramp is 360 newtons. Because the refrigerator is 100 kilograms, you have an acceleration of 360 N/100 kg = 3.6 m/s2, which acts along the entire 3.0-meter ramp. You can calculate the final speed of the refrigerator at the bottom of the ramp this way:

image7.png

Plugging in the numbers, you get

image8.png

The final speed of the refrigerator when it starts traveling along the street toward the parked car is about 4.6 meters per second.

Now you need to calculate how far the refrigerator is going to go. Because it’s traveling along the pavement now, you need to factor in the force due to friction. Gravity will no longer accelerate the object because the street is flat. Sooner or later, the refrigerator will come to a stop. But how close will it come to a car that’s parked in the street 7.2 meters away? As usual, your first calculation is the force acting on the object. In this case, you figure the magnitude of the force due to friction:

image9.png

Because the refrigerator is moving along a horizontal surface, the normal force, FN, is simply the weight of the refrigerator, mg, which means the force of friction is

image10.png

Plugging in the data gives you

image11.png

A force of 150 newtons acts to stop the sliding refrigerator that’s now terrorizing the neighborhood. So how far will it travel before it comes to rest? If you take the refrigerator to be moving horizontally in the positive direction, then because the force is acting in the opposite direction, its horizontal component is negative. Because of Newton’s second law, the acceleration is also negative and is given by

image12.png

You can find the distance through the equation

image13.png

The distance the refrigerator slides is

image14.png

In this case, you want the final velocity, vf, to be zero, because you need to know where the refrigerator will stop. Therefore, this equation breaks down to

image15.png

Whew! The refrigerator slides only 7.1 meters, and the car is 7.2 meters away. With the pressure off, you watch the show as your panic-stricken friends hurtle after the refrigerator, only to see it come to a stop right before hitting the car — just as you expected.

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