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 at which mass m must move for mass M to stay at rest, we can start by considering the forces acting on the system.
Let's assume that the speed of mass m is v.
For mass M to stay at rest, the centripetal force required to keep it in circular motion should be balanced by the tension in the cord.
The centripetal force acting on mass M is given by:
F = (M)(a)
where M is the mass of object M and a is the centripetal acceleration.
Since mass M is at rest, its acceleration is zero. Therefore, the centripetal force is also zero.
Now, let's consider the forces acting on mass m. The centripetal force acting on mass m is provided by the tension in the cord. The magnitude of the tension should be equal to the centripetal force required for mass m to move in a circle of radius 0.420 m at a constant speed v.
The centripetal force acting on mass m is given by:
F = (m)(a)
where m is the mass of object m and a is the centripetal acceleration.
The centripetal acceleration is given by:
a = (v^2) / r
where v is the speed of mass m and r is the radius of the circular path.
Since the system is frictionless, the only force acting on mass m is the tension in the cord. Therefore, the tension in the cord must provide the necessary centripetal force for mass m.
Setting the centripetal force equal to the tension, we have:
(m)(a) = T
Substituting the expression for centripetal acceleration, we get:
(m)(v^2 / r) = T
Now, we can solve this equation for v to find the speed at which mass m must move:
v^2 = (T * r) / m
v = sqrt((T * r) / m)
Now, we need to find the tension in the cord. Since the tension is the same throughout the cord, we can consider the forces acting on mass M to find the tension.
The net force acting on mass M is given by:
F_net = m * g - T
where m is the mass of object M, g is the acceleration due to gravity, and T is the tension in the cord.
Since mass M is at rest, the net force on it should be zero.
Therefore, we have:
m * g - T = 0
T = m * g
Substituting this expression for T into the equation for v, we get:
v = sqrt(((m * g) * r) / m)
Simplifying, we find:
v = sqrt(g * r)
Now, we can plug in the values given in the problem to calculate the speed v:
g = 9.8 m/s^2
r = 0.420 m
Plugging these values into the equation, we have:
v = sqrt(9.8 * 0.420)
Calculating this, we find:
v ≈ 1.91 m/s
Therefore, mass m must move with a speed of approximately 1.91 m/s for mass M to stay at rest.