A certain light truck can go around a flat curve having a radius of 150 m with a maximum speed of 34.0 m/s. With what maximum speed can it go around a curve having a radius of 60.0 m?
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To determine the maximum speed at which the light truck can go around a curve with a smaller radius, we can use the principle of conservation of energy.
The centripetal force required to keep the truck moving in a circular path is provided by the frictional force between the tires and the road. The frictional force can be calculated using the equation:
F = m * (v^2 / r),
where F is the frictional force, m is the mass of the truck, v is the velocity, and r is the radius of the curve.
In this case, we're assuming that the mass of the truck remains constant. Therefore, we can rewrite the equation as:
F = k * v^2,
where k is a constant determined by the mass and the radius of the curve.
Since we're interested in the maximum speed, we need to find the value of v at which the frictional force is at its maximum. The maximum frictional force occurs when the tires are on the verge of slipping, which is represented by the friction coefficient μ multiplied by the normal force between the tires and the road.
F_max = μ * N,
where μ is the friction coefficient and N is the normal force.
In this case, the maximum frictional force is equal to the centripetal force required to keep the truck moving in a circular path, so we can equate the two equations:
k * v^2 = μ * N.
Since the mass of the truck remains constant, the normal force remains the same for both curves. Therefore, we can simplify the equation to:
v^2 = constant,
meaning the square of the velocity is constant for curves with different radii.
To find the maximum velocity for the smaller curve, we can use the constant value from the larger curve:
v_small^2 = v_large^2.
Substituting the given values for the larger curve:
v_small^2 = (34.0 m/s)^2.
Simplifying, we find:
v_small = sqrt((34.0 m/s)^2) = 34.0 m/s.
Thus, the maximum speed at which the light truck can go around a curve with a radius of 60.0 m is 34.0 m/s.