AP Physics

Unit 2 - Linear Forces

Intermediate

Mathematical

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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?

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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=μsNf_s = \mu_s N The maximum static friction force (fsf_s) is equal to the coefficient of static friction (μs\mu_s) times the normal force (NN).
2 N=mgcos(θ)N = mg \cos(\theta) The normal force is the component of the gravitational force perpendicular to the slope, where mm is the mass of the car, gg is the acceleration due to gravity, and θ\theta is the angle of the hill.
3 fs=mgsin(θ)f_s = mg \sin(\theta) The force down the slope is the component of gravity parallel to the slope.
4 Set fsf_s from step 1 equal to fsf_s from step 3. <br> μsmgcos(θ)=mgsin(θ)\mu_s mg \cos(\theta) = mg \sin(\theta) Equate the maximum static friction force to the gravitational force down the slope.
5 Simplify and solve for θ\theta. <br> μscos(θ)=sin(θ)tan(θ)=μs\mu_s \cos(\theta) = \sin(\theta) \Rightarrow \tan(\theta) = \mu_s <br> θ=arctan(μs)\theta = \arctan(\mu_s) Divide both sides by mgcos(θ)mg \cos(\theta) and rearrange to solve for θ\theta.
6 θ=arctan(0.85)\theta = \arctan(0.85) Substitute μs=0.85\mu_s = 0.85 into the equation.

Now, calculate the angle θ\theta:

Step Result
7 θ40.36 \boxed{\theta \approx 40.36^\circ}

The maximum angle of a hill on which a car can be parked without sliding, given the coefficient of static friction, is approximately 40.3640.36^\circ.

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