If a curve with a radius of 88m is perfectly banked for a car traveling 75km/h, what must be the coefficient of static friction for a car not skid when traveling at 95 km/h?
You forgot the normal force due to centripetal force.
mg*sinTheta+mu*mg*cosTheta+mu*v^2/r*sinTheta=mv^2/r * cosTheta
m cancels out so the equilibrium is stated in acceleration.
mu= ((v^2/r)*cosTheta-g*sinTheta)/(g*cosTheta+(v^2/r)*sinTheta)
To answer this question, we can start by analyzing the forces acting on the car as it travels around the curve. The two main forces involved are the gravitational force (mg) and the static friction force (f).
To prevent the car from skidding, the static friction force must provide the necessary centripetal force to keep the car moving in a circular path. The centripetal force is given by the equation:
F = (mv^2) / r
where F is the centripetal force, m is the mass of the car, v is its velocity, and r is the radius of the curve.
In this case, we need to find the coefficient of static friction (μ) required for the car to not skid when traveling at 95 km/h. The maximum static friction force can be calculated using:
f = μN
where N is the normal force exerted by the road on the car. The normal force is equal to the weight of the car, which can be calculated as:
N = mg
Now, let's plug in the given values and calculate the required coefficient of static friction.
First, we need to convert the velocity from km/h to m/s:
v = 95 km/h * (1000 m / 3600 s) = 26.39 m/s
Next, we plug in the known values for the radius and velocity into the centripetal force equation:
F = (m * v^2) / r
F = (m * 26.39^2) / 88
Now, we know that the maximum static friction force required to prevent skidding is equal to the centripetal force. Therefore:
f = F
Therefore, we can write:
f = (m * 26.39^2) / 88
Finally, we can substitute the equation for the maximum static friction force (f) into the equation for the coefficient of static friction (μ):
(μ * mg) = (m * 26.39^2) / 88
Simplifying the equation:
μ = (26.39^2) / (88 * g)
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, you can calculate the coefficient of static friction using the given equation and the value of g.
Can you show me step by step with the numbers in there? I am confused..
change velocities to m/s
Then, sketch the bank and force vectors parallel to the bank.
1) mg*sinTheta=mv^2/r * cosTheta
where v=75km/hr in m/s
solve for tan theta (sin/cos).
verify that. Now, you increase speed, same angle.
mg*sinTheta+mu*mg*cosTheta=mv^2/r * cosTheta
so find mu. How is the easy way? divid e by cos theta , you know tan theta.
v= new speed (95 in m/s), solve for mu.