A figure skater pushes off from rest and glides along a circular path, the radius of which is 3 times her height. What must be her minimum initial velocity in order to come full circle without having to push off a second time?
To find the minimum initial velocity required for the figure skater to come full circle without having to push off a second time, we can use the concept of centripetal force.
The centripetal force required to keep an object moving in a circular path is given by the equation:
F = (mv^2) / r
Where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.
In this case, we can assume that the figure skater's mass is constant and does not affect the minimum initial velocity required. Therefore, we can simplify the equation to:
F = (m * v^2) / (3h)
Where h is the height of the figure skater, and r (radius) is equal to 3h.
Since the figure skater wants to complete a full circle without having to push off a second time, the centripetal force must be equal to the gravitational force acting on her. This is given by:
F = m * g
Where g is the acceleration due to gravity.
Setting these two equations equal to each other, we have:
(m * v^2) / (3h) = m * g
Simplifying:
v^2 = 3gh
Finally, taking the square root of both sides, we get:
v = √(3gh)
So, the minimum initial velocity (v) that the figure skater needs to ensure she completes a full circle without pushing off a second time is equal to the square root of 3gh, where g is the acceleration due to gravity and h is her height.