A mass m = 0.225 kg on a frictionless table is attached to a hanging mass M = 0.330 kg by a cord through a hole in the table (see the figure). Find the speed with which m must move in order for M to stay at rest if r = 27 cm.
To find the speed at which mass m must move in order for mass M to stay at rest, we can use the principle of conservation of momentum.
The basic idea is that the momentum before the collision should be equal to the momentum after the collision. In this case, the collision happens when mass m moves downwards and pulls mass M upwards.
The momentum of an object is given by the product of its mass and velocity (p = mv). Since mass M is at rest, its velocity is 0, so its momentum is also 0.
Before the collision, only mass m is moving, so its momentum is given by p = mv, where v is the velocity of mass m.
During the collision, the cord is assumed to be inextensible, so the displacement of the system is equal to the length of the cord pulled through the hole in the table. This length is given by r, the radius of the circular motion of mass m.
We can use the fact that the displacement of an object moving in a circle is equal to the product of the radius and the angle in radians. Since mass m moves in a circle of radius r, the angle through which it moves is given by θ = 2π radians.
Therefore, the displacement of mass m is s = rθ = r(2π). Using this displacement and the time taken to cover this distance (which is the same as the time taken for mass M to stay at rest), we can find the velocity v of mass m.
The time taken can be found using the equation v = s/t, where v is the velocity of mass m, s is the displacement, and t is the time taken.
Substituting the expressions for s and t, we have v = r(2π) / t.
Now we have the velocity v of mass m. Since mass M and mass m are connected by the cord, the velocity of mass M must also be v.
Finally, the speed with which mass m must move in order for mass M to stay at rest is the magnitude of the velocity v, which is given by the absolute value of v.
Therefore, the speed with which mass m must move is v = |r(2π) / t|.
To find the time t, we can use the equation of the system's motion under gravity, where the acceleration due to gravity is g.
The force acting on mass m is its weight m * g, which is balanced by the tension in the cord, T. This tension is also equal to the force that keeps mass M at rest.
Using Newton's second law, F = ma, where F is the net force on mass m and a is its acceleration, we have m * g = m * a.
Since the tension in the cord is also equal to the force on mass M, we have T = M * g.
Since mass M is at rest, the force on it is zero, so we have T = 0.
We can now rewrite the equation for the net force on mass m as m * g = m * a.
The acceleration a of mass m can be found using the equation of motion s = ut + 0.5 * a * t^2, where s is the displacement, u is the initial velocity, a is acceleration, and t is time.
In this case, the initial velocity u of mass m is 0, since it starts from rest.
Since the displacement s is r(2π), we have r(2π) = 0.5 * a * t^2.
From this equation, we can solve for t.
First, we can rearrange the equation to get t^2 = 2 * r(2π) / a.
Then, we can substitute the value of a obtained from the equation m * g = m * a:
t^2 = 2 * r(2π) / (m * g).
Finally, we can take the square root of both sides to get t:
t = sqrt(2 * r(2π) / (m * g)).
Now that we have the time t, we can substitute it back into the expression for velocity v to find the speed with which mass m must move:
v = |r(2π) / t|.
Hmmm. I assume M is rotating.
Mv^2/r=mg
solve for v.