The rated speed of the highway curve of 200 ft radius is 30 mph. If the coefficient of friction between the tires and the road is 0.6, what is the maximum speed at which a car can round the curve without skidding?
show solution step by step
show FBD (if possible)
thanks
Thanks for posting the questions. However, can you please show any work you have done for any of these? Thank you.
I have 20 problems here, i've only finish 7 out of 20,, this giving me so much headache... hoohoohoo
Okay, but can you post any work you’ve done so far.
You need a force to balance mass times centripetal acceleration.
That force is provided by the weight times the coefficient of friction.
To find the maximum speed at which a car can round the curve without skidding, we can use the concept of centripetal force.
The centripetal force is the force required to keep an object moving in a curved path. In this case, it is provided by the friction between the tires and the road.
The formula for centripetal force is:
Fc = (mv^2) / r
where m is the mass of the car, v is the speed of the car, and r is the radius of the curve.
In this case, we are given that the radius of the curve is 200 ft and the coefficient of friction is 0.6. Let's assume the mass of the car is 1,000 kg.
Step 1: Convert the radius to meters.
1 ft is approximately equal to 0.305 meters. So, the radius of the curve is 200 ft * 0.305 m/ft = 61 m.
Step 2: Plug the values into the centripetal force formula.
Fc = (mv^2) / r
Fc = (1000 kg) * v^2 / 61 m
Step 3: Substitute the coefficient of friction into the equation.
The frictional force is given by Ff = μN, where μ is the coefficient of friction and N is the normal force. In this case, the normal force is equal to the weight of the car, which is equal to mg.
Therefore, the frictional force Ff = μmg.
Since the frictional force is the centripetal force, we have Ff = Fc.
Therefore, μmg = mv^2 / r
Step 4: Solve for the maximum speed.
μmg = mv^2 / r
v^2 = r * μmg / m
v = sqrt( r * μ * g)
Substituting the values into the equation:
v = sqrt(61 m * 0.6 * 9.8 m/s^2)
Simplifying the expression, we get:
v ≈ sqrt(359.28) ≈ 18.96 m/s
To convert this to miles per hour, we multiply by the conversion factor 2.237:
v ≈ 18.96 m/s * 2.237 ≈ 42.41 mph
Therefore, the maximum speed at which a car can round the curve without skidding is approximately 42.41 mph.
Unfortunately, it is not possible to show a Free Body Diagram (FBD) for this particular problem since we are dealing with forces in the radial direction and not the traditional horizontal or vertical directions.