A certain light truck can go around a flat curve having a radius of 150 m with a maximum speed of 36.0 m/s. With what maximum speed can it go around a curve having a radius of 83.5 m?
To solve this problem, we can use the concept of centripetal force. The centripetal force is the force exerted on an object moving in a circular path, directed towards the center of the circle. It can be calculated using the equation:
Fc = (mv^2) / r
Where Fc is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.
In this case, we are assuming that the mass of the truck remains constant. So we can ignore it in our calculations.
For the first scenario, where the radius is 150 m and the maximum speed is 36.0 m/s, we can calculate the centripetal force using the given values:
Fc1 = (m * v1^2) / r1
Now, for the second scenario, where the radius is 83.5 m and we need to find the maximum speed, let's label it as v2. We can calculate the centripetal force for this scenario as well:
Fc2 = (m * v2^2) / r2
Since the mass of the truck is the same in both scenarios, we can set the centripetal forces equal to each other and solve for v2:
(m * v1^2) / r1 = (m * v2^2) / r2
We can cancel out the mass, and rearrange the equation to solve for v2:
v2^2 = (v1^2 * r2) / r1
Finally, take the square root of both sides to find the maximum speed (v2) for the second scenario:
v2 = sqrt((v1^2 * r2) / r1)
Now, plug in the values:
v1 = 36.0 m/s
r1 = 150 m
r2 = 83.5 m
v2 = sqrt((36.0^2 * 83.5) / 150)
After performing the calculations, we find that the maximum speed (v2) the truck can go around the curve with a radius of 83.5 m is approximately 26.12 m/s.