3. A car with a mass of 1.8 × 103 kg is parked on a hill in San Francisco. Suppose the hill makes a

Question

15.0° incline with the horizontal. If the frictional force required to keep the car from sliding down the
hill is 1.25 × 104 N, what is the coefficient of static friction between the pavement and the car’s tires?

Answer 1

To find the coefficient of static friction (\( \mu_s \)) required to keep the car from sliding down the hill, we first need to analyze the forces acting on the car parked on the incline.

1. Identify the forces:

   • The gravitational force acting on the car is given by: \[ F_g = mg \] where:
     • \( m = 1.8 \times 10^3 , \text{kg} \) (mass of the car)
     • \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity)

2. Calculate the gravitational force: \[ F_g = (1.8 \times 10^3 , \text{kg}) \times (9.81 , \text{m/s}^2) \approx 17658 , \text{N} \]

3. Determine the components of the gravitational force on the incline:

   • The component of gravitational force acting down the incline can be calculated using: \[ F_{\text{down}} = F_g \sin(\theta) \] where \( \theta = 15.0^\circ \).

4. Calculate \( F_{\text{down}} \): \[ F_{\text{down}} = 17658 , \text{N} \cdot \sin(15.0^\circ) \approx 17658 , \text{N} \cdot 0.2588 \approx 4565 , \text{N} \]

5. Frictional force condition:

   • The frictional force (\( F_{\text{friction}} \)) is required to balance the component of gravitational force acting down the incline to prevent the car from sliding: \[ F_{\text{friction}} = F_{\text{down}} = 1.25 \times 10^4 , \text{N} \]

6. Relate the frictional force to the normal force:

   • The maximum static friction force is given by: \[ F_{\text{friction}} = \mu_s F_n \] where \( F_n \) is the normal force.
   • The normal force \( F_n \) can be calculated as: \[ F_n = F_g \cos(\theta) = mg \cos(\theta) \]

7. Calculate the normal force: \[ F_n = (1.8 \times 10^3 , \text{kg}) \cdot (9.81 , \text{m/s}^2) \cdot \cos(15.0^\circ) \] \[ F_n \approx 17658 , \text{N} \cdot 0.9659 \approx 17078 , \text{N} \]

8. Calculate the coefficient of static friction:

   • Substitute the known values into the friction equation: \[ \mu_s = \frac{F_{\text{friction}}}{F_n} = \frac{1.25 \times 10^4 , \text{N}}{17078 , \text{N}} \]
   • Calculating \( \mu_s \): \[ \mu_s \approx \frac{12500}{17078} \approx 0.731 \]

Thus, the coefficient of static friction between the pavement and the car's tires is approximately \( \mu_s \approx 0.731 \).