A motorcycle stunt rider wants to do a loop-the-loop within a vertical circular track. If the radius of the circular track is 10.0 m, what minimum speed must the motorcyclist maintain to stay on the track?
Suppose the radius of the track was doubled. By what factor will the motorcyclist need to increase her speed to loop-the-loop on the new track?
To determine the minimum speed the motorcyclist must maintain to stay on the track, we can use the concept of centripetal force.
The centripetal force required to keep an object moving in a circle is given by the formula:
F = m * (v^2 / r),
where F 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 track.
In this case, the motorcyclist will need to exert a force equal to their weight to stay on the loop. Therefore, the centripetal force can be expressed as:
F = m * g,
where g is the acceleration due to gravity.
Setting these two expressions equal to each other, we have:
m * g = m * (v^2 / r).
The mass of the motorcyclist cancels out, leaving us with:
g = v^2 / r.
Now, let's solve for v to find the minimum speed the motorcyclist must maintain:
v^2 = g * r.
Taking the square root of both sides, we get:
v = sqrt(g * r).
Substituting the given values of the radius (r = 10.0 m) and acceleration due to gravity (g = 9.8 m/s^2), we can calculate the minimum speed:
v = sqrt(9.8 * 10.0) = sqrt(98) ≈ 9.90 m/s.
Therefore, the motorcyclist must maintain a minimum speed of approximately 9.90 m/s to stay on the track.
Now, let's consider the second part of the question where the radius of the track is doubled.
If the radius is doubled to 2 * 10.0 m = 20.0 m, we need to determine the factor by which the motorcyclist must increase her speed to loop-the-loop on the new track.
Using the same formula as before:
v = sqrt(g * r),
we can substitute the new radius (r = 20.0 m) and solve for v:
v = sqrt(9.8 * 20.0) = sqrt(196) = 14.0 m/s.
Comparing this to the original speed of 9.90 m/s, we can calculate the factor by which the speed needs to be increased:
Factor = New Speed / Original Speed = 14.0 m/s / 9.90 m/s ≈ 1.41.
Therefore, the motorcyclist would need to increase her speed by a factor of approximately 1.41 to loop-the-loop on the new track with a doubled radius.