A 3.50 kg mass, m, on a frictionless table is moving in a circle with radius 0.420 m at a constant speed. m is attached to a 6.40 kg mass, M, by a cord through a hole in the table. Find the speed with which m must move for M to stay at rest.
To find the speed with which the mass m must move for M to stay at rest, 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:
Fc = (mv^2) / r
Where Fc is the centripetal force, m is the mass of the moving object, v is its velocity, and r is the radius of the circular path.
In this case, since mass M must stay at rest, the tension in the cord pulling it towards the center must be equal to its weight:
T = Mg
Where T is the tension, M is the mass of object M, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
We can equate these two equations:
T = Fc
Mg = (mv^2) / r
Now, we can solve for v.
First, isolate v^2 by multiplying both sides of the equation by r:
Mg * r = mv^2
Next, divide both sides of the equation by m:
v^2 = (Mg * r) / m
Finally, take the square root of both sides to find v:
v = sqrt((Mg * r) / m)
Substituting the given values:
M = 6.40 kg, g = 9.8 m/s^2, r = 0.420 m, and m = 3.50 kg
v = sqrt((6.40 kg * 9.8 m/s^2 * 0.420 m) / 3.50 kg)
v ≈ 3.09 m/s
Therefore, the speed with which mass m must move for mass M to stay at rest is approximately 3.09 m/s.