An object, previously at rest, slides #3 m# down a ramp, with an incline of #pi/12 #, and then slides horizontally on the floor for another #15 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
WARNING: LONG-ish ANSWER!
We're asked to find the coefficient of kinetic friction,
We'll split this problem into two parts: the first part is where the object is sliding down the incline, and the second part is where it is sliding across the floor.
The only two forces acting on the object as it slides down the ramp are The gravitational force (its weight; acting down the ramp)
The kinetic friction force (acting up the ramp because it opposes motion) The expression for the coefficient of kinetic friction where The expression for the net horizontal force, which I'll call Since And since the normal force Or Using Newton's second law, we can find the expression for the acceleration What we can now do is apply a constant-acceleration equation to find the final velocity as it exits the ramp, which we'll call where Plugging in these values: This velocity is also the initial velocity of the motion along the floor. As the object slides across the floor, the plane is perfectly horizontal, so the normal force The only horizontal force acting on the object is the retarding kinetic friction force (which is different than the first one). The net horizontal force on the object on the floor, which we'll call (the friction force is negative because it opposes the object's motion) Using Newton's second law again, we can find the floor acceleration We can now use the same constant-acceleration equation as before, with only a slight difference in the variables: where this time Plugging in these values: Rearranging gives At this point, we're just solving for Divide both sides by Distribute: Now, we can divide all terms by Rearrange: Finally, swap The question gives us Plugging these in: Notice how the coefficient doesn't depend on the mass
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The material's kinetic friction coefficient is approximately 0.12.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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