How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85m at a speed of 96km/hr?
How do I figure this out if no mass is given?
To figure out the necessary coefficient of static friction between the tires and the road in order for the car to navigate the curve at the given speed, we can use the concept of centripetal force.
First, let's break down the problem into steps:
Step 1: Convert the given speed from km/hr to m/s.
- To do this, we can use the conversion factor: 1 km/hr = 1000 m/3600 s.
- So, 96 km/hr = (96 × 1000) m/3600 s = 26.67 m/s (rounded to two decimal places).
Step 2: Determine the centripetal force required to keep the car moving in a circular path.
- Centripetal force is given by the formula: F = (m × v^2) / r, where F is the centripetal force, m is the mass of the car, v is the velocity of the car, and r is the radius of the curvature.
- Since the mass of the car is not given, we can cancel it out from the equation because it appears on both sides.
- So, F = (v^2) / r.
Step 3: Find the maximum static friction force.
- The maximum static friction force (F_max) is given by the formula: F_max = μ_s × N, where μ_s is the coefficient of static friction and N is the normal force.
- In this case, the normal force exerted by the ground on the car is equal to the weight of the car (mg), where g is the acceleration due to gravity.
- Since the mass of the car is not given, we can again cancel it out from the equation because it appears on both sides.
- So, F_max = μ_s × N = μ_s × mg.
Step 4: Equate the centripetal force to the maximum static friction force.
- Set the centripetal force (F) equal to the maximum static friction force (F_max) and solve for μ_s.
- We have: (v^2) / r = μ_s × mg.
Step 5: Rearrange the equation to solve for μ_s.
- Divide both sides of the equation by g: (v^2) / (rg) = μ_s.
- Now substitute the given values into the equation to find μ_s.
- v = 26.67 m/s (from Step 1).
- r = 85 m (given radius).
Finally, plug in the values and calculate:
- μ_s = (26.67^2) / (85 × 9.8) = 0.942 (rounded to three decimal places).
Therefore, to round the level curve at a speed of 96 km/hr, the coefficient of static friction between the tires and the road must be at least 0.942.