A circular off ramp has a radius of 74 m and a posted speed limit of 50km/h. If the road is horizontal, what is the minimum coefficient of friction needed?
frictionforce=centripetal force
mu*mg=m*v^2/r
v has to be in m/s
To find the minimum coefficient of friction needed, we need to consider the forces acting on the vehicle at the maximum speed allowed on the circular off ramp.
The maximum speed at the off ramp is given as 50 km/h. We need to convert this speed into meters per second (m/s) for ease of calculations.
1 km/h is equal to 1000 meters per hour. Since there are 60 minutes in an hour and 60 seconds in a minute, we can convert kilometers per hour to meters per second as follows:
50 km/h = (50 * 1000) meters/3600 seconds = 13.89 meters/second (approx)
Now, let's analyze the forces acting on the vehicle at this speed. At this speed, the centripetal force necessary to keep the vehicle moving in a circular path is provided by the frictional force between the tires and the road.
The centripetal force (F) is given by the equation:
F = (mass of the vehicle) * (centripetal acceleration)
The centripetal acceleration (a) can be calculated using the formula:
a = (velocity^2) / radius
Plugging in the given values:
a = (13.89^2) / 74 = 2.6 m/s^2 (approx)
Now, dividing the centripetal force (F) by the gravitational force (mg) gives us the coefficient of friction (μ).
F = μ * (mass of the vehicle) * (gravitational acceleration)
μ = F / (mass of the vehicle) * (gravitational acceleration)
The gravitational acceleration (g) is approximately 9.8 m/s^2.
Let's assume the mass of the vehicle to be 1000 kg.
μ = (2.6 * 1000) / (1000 * 9.8) = 0.265
Therefore, the minimum coefficient of friction needed for the vehicle to safely navigate the circular off ramp at the posted speed limit of 50 km/h is approximately 0.265.