On a wet racetrack, the maximum safe speed, to avoid sliding on a level circular curve with a radius of curvature of 56 m, is 73 km/hr. What is the coefficient of friction between the car's tires and the track surface?
First convert 73 km/hr to
Vmax = 20.28 m/s
Then solve
M*g*u = M*V^2/R
or, since the M cancels out,
u = Vmax^2/(R*g)
u is the coefficient of friction that you want.
To find the coefficient of friction between the car's tires and the track surface, we can use the centripetal force equation:
Fc = (m * v^2) / r
Where:
- Fc is the centripetal force,
- m is the mass of the car,
- v is the velocity of the car, and
- r is the radius of curvature.
In this case, we need to convert the maximum safe speed from km/hr to m/s:
v = (73 km/hr) * (1000 m/km) / (3600 s/hr) = 73 * 1000 / 3600 m/s ≈ 20.2778 m/s
Now, we can rearrange the formula to solve for the coefficient of friction (μ):
Fc = μ * (m * g)
Simplifying the equation:
μ = (Fc) / (m * g)
Where:
- Fc is the centripetal force,
- m is the mass of the car, and
- g is the acceleration due to gravity (approximated as 9.8 m/s^2).
We can substitute the centripetal force equation into the coefficient of friction equation:
μ = [(m * v^2) / r] / (m * g)
Simplifying further:
μ = (v^2) / (r * g)
Substituting the given values:
μ = (20.2778 m/s)^2 / (56 m * 9.8 m/s^2)
Calculating:
μ ≈ 0.079
Therefore, the coefficient of friction between the car's tires and the track surface is approximately 0.079.
To find the coefficient of friction between the car's tires and the track surface, we can start by analyzing the forces acting on the car.
In this case, the maximum safe speed to avoid sliding can be achieved by ensuring that the centripetal force acting on the car is equal to the maximum force of static friction between the tires and the wet track surface.
The centripetal force is given by the equation:
Fc = mv² / r
Where:
- Fc is the centripetal force
- m is the mass of the car
- v is the velocity of the car
- r is the radius of curvature
At the maximum safe speed, the centripetal force is equal to the maximum force of friction between the tires and the track surface. Therefore, we can equate these two forces:
mv² / r = μN
Where:
- μ is the coefficient of friction
- N is the normal force (equal to the weight of the car)
To find the normal force N, we can use the equation:
N = mg
Where:
- m is the mass of the car
- g is the acceleration due to gravity (approximately 9.8 m/s²)
We need to convert the given maximum safe speed from km/hr to m/s:
73 km/hr = 73 * 1000 m / 3600 s ≈ 20.28 m/s
Now, we can plug in the values to solve for the coefficient of friction:
m * (20.28 m/s)² / 56 m = μ * m * 9.8 m/s²
Simplifying the equation:
(20.28 m/s)² / 56 m = μ * 9.8 m/s²
407.5224 m²/s² / 56 m = 9.8 μ
7.2761 m/s² = 9.8 μ
Now, divide both sides by 9.8 m/s² to solve for μ:
μ = 7.2761 m/s² / 9.8 m/s² ≈ 0.74
Therefore, the coefficient of friction between the car's tires and the track surface is approximately 0.74.