A car is moving on a banked curve with a radius of 50 meters. The coefficient of friction between the tires and the road is 0.4. Calculate the maximum speed the car can safely travel around the curve without skidding.
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To calculate the maximum speed the car can safely travel around the curve without skidding, we need to consider the forces acting on the car.
The two main forces involved are the gravitational force (mg) acting vertically downwards and the frictional force (μN) acting horizontally towards the center of the curve.
Since the car is not skidding, the frictional force provides the required centripetal force to keep the car moving in a circle.
The maximum frictional force (Ffmax) can be calculated using the coefficient of friction (μ) and the normal force (N):
Ffmax = μN
The normal force (N) is equal to the weight of the car (mg):
N = mg
Substituting N into the equation for Ffmax:
Ffmax = μmg
The centripetal force (Fc) required to keep the car moving in a circle is given by:
Fc = mv² / r
Where m is the mass of the car, v is the velocity, and r is the radius of the curve.
When the car is at the maximum speed without skidding, the frictional force (Ffmax) is equal to the centripetal force (Fc):
Ffmax = Fc
μmg = mv² / r
To find the maximum speed (vmax), we rearrange the equation:
vmax = √(μrg)
Now we can substitute the given values into the equation:
μ = 0.4 (coefficient of friction)
r = 50 meters (radius)
g = 9.8 m/s² (acceleration due to gravity)
vmax = √(0.4 * 50 * 9.8)
Calculating this gives us:
vmax ≈ 14.0 m/s
Therefore, the maximum speed the car can safely travel around the curve without skidding is approximately 14.0 m/s.
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