The coefficient of static friction between hard rubber and normal street pavement is about 0.85. On how steep a hill (maximum angle) can you leave a car parked?
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The coefficient of static friction between hard rubber and normal street pavement is about 0.85. On how steep a hill (maximum angle) can you leave a car parked?
Objective: Determine the maximum angle of a hill on which a car can be parked without sliding, given the coefficient of static friction between the tires and the pavement.
Step | Formula Derivation | Reasoning |
---|---|---|
1 | fs=μsN | The maximum static friction force (fs) is equal to the coefficient of static friction (μs) times the normal force (N). |
2 | N=mgcos(θ) | The normal force is the component of the gravitational force perpendicular to the slope, where m is the mass of the car, g is the acceleration due to gravity, and θ is the angle of the hill. |
3 | fs=mgsin(θ) | The force down the slope is the component of gravity parallel to the slope. |
4 | Set fs from step 1 equal to fs from step 3. <br> μsmgcos(θ)=mgsin(θ) | Equate the maximum static friction force to the gravitational force down the slope. |
5 | Simplify and solve for θ. <br> μscos(θ)=sin(θ)⇒tan(θ)=μs <br> θ=arctan(μs) | Divide both sides by mgcos(θ) and rearrange to solve for θ. |
6 | θ=arctan(0.85) | Substitute μs=0.85 into the equation. |
Now, calculate the angle θ:
Step | Result |
---|---|
7 | θ≈40.36∘ |
The maximum angle of a hill on which a car can be parked without sliding, given the coefficient of static friction, is approximately 40.36∘.
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~ 40°
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