A car travels around a circular track (4 km diameter). Its tires have frictional coefficients: mS = 0.3 & mK = 0.2. If the track is flat, how fast can the car travel around the track and still maintain control?
To calculate the maximum speed at which the car can travel around the track and maintain control, we need to consider the frictional forces acting on the tires. The two types of frictional coefficients provided - mS (static friction coefficient) and mK (kinetic friction coefficient) - are important in determining this maximum speed.
Here's how you can calculate the maximum speed:
1. Determine the maximum frictional force available:
The maximum static frictional force (Fmax) can be calculated using the formula:
Fmax = mS * Normal force,
where the normal force is the force exerted by the track perpendicular to the tires. In this case, it is equal to the weight of the car, which can be calculated as:
Weight = mass * gravitational acceleration.
2. Calculate the actual frictional force acting on the tires:
The actual frictional force can be calculated using the formula:
F = mK * Normal force.
3. Equate the maximum frictional force to the actual frictional force:
Set Fmax = F and solve for the normal force.
4. Calculate the maximum speed using the formula:
Maximum speed = square root of (μ * g * r),
where μ is the coefficient of kinetic friction (mK), g is the gravitational acceleration, and r is the radius of the circular track (half the diameter).
Here's an example calculation:
Given:
Diameter of the circular track = 4 km
mS = 0.3
mK = 0.2
1. Calculate the maximum frictional force available:
The weight of the car can be calculated as:
Weight = mass * gravitational acceleration.
Assume the mass of the car is 1000 kg.
Weight = 1000 kg * 9.8 m/s^2 = 9800 N.
The maximum static frictional force (Fmax) can be calculated as:
Fmax = mS * Normal force = 0.3 * 9800 N = 2940 N.
2. Calculate the actual frictional force acting on the tires:
The actual frictional force (F) can be calculated as:
F = mK * Normal force = 0.2 * Normal force.
3. Equate the maximum frictional force to the actual frictional force:
Fmax = F, so 2940 N = 0.2 * Normal force.
Solving for Normal force, we get: Normal force = 2940 N / 0.2 = 14700 N.
4. Calculate the maximum speed:
The radius of the track is half the diameter, so r = 4 km / 2 = 2 km = 2000 m.
Using the formula:
Maximum speed = square root of (mK * g * r),
Maximum speed = square root of (0.2 * 9.8 m/s^2 * 2000 m) = square root of (3920 m^2/s^2) = 62.5 m/s.
Therefore, the car can travel at a maximum speed of approximately 62.5 m/s around the circular track and still maintain control.