Mass m1 on the frictionless table of the figure below is connected by a string through a hole in the table to a hanging mass m2. With what speed must m1 rotate in a circle of radius r if m2 is to remain hanging at rest? (Use any variable or symbol stated above along with the following as necessary: g.)
Fm1=(m1v^2)/r
Fm2=mg=9.8m2
Rewrite Fm1=(m1v^2)/r as v= sqrt Fr/sqrt m1
Plug Fm2 into the equation to get: sqrt 9.8m2r/sqrt m1
v=sqrt (m2(g)(r))/m1
m2 g = m1 v^2/r
Well, the first thing I have to say is, this table must be really bored if it wants to see m1 rotating in a circle. Maybe it's just looking for some entertainment!
But let's get serious for a second. To keep m2 hanging at rest, we need to consider the tension in the string. The tension in the string is providing the necessary centripetal force to keep m2 from falling. So we can equate the tension force and the gravitational force acting on m2.
The tension force T in the string is given by T = m2 * g, where g is the acceleration due to gravity.
Now, the centripetal force acting on m1 is given by F = m1 * v^2 / r, where v is the speed of m1. Since there is no friction, the tension in the string is the centripetal force, so we have T = m1 * v^2 / r.
Setting these two equations equal to each other, we get m2 * g = m1 * v^2 / r.
Solving for v, we have v = sqrt((m2 * g * r) / m1).
So, to keep m2 hanging at rest, m1 must rotate at a speed of sqrt((m2 * g * r) / m1).
I hope that helps, and that this table gets its share of laughter from m1's spinning antics!
To find the speed at which mass m1 must rotate in a circle of radius r for mass m2 to remain hanging at rest, we need to consider the forces acting on m2.
There are two forces acting on m2:
1. The gravitational force pulling m2 downwards with a magnitude of m2 * g (where g is the acceleration due to gravity).
2. The tension force in the string caused by the rotation of m1.
Since m2 is hanging at rest, the net force on it must be zero. This means that the tension in the string must be equal to the gravitational force acting on m2.
Tension = m2 * g
The tension in the string is also related to the centripetal force required to keep m1 moving in a circle of radius r. The tension in the string is equal to the centripetal force.
Centripetal Force = Tension = m1 * (v^2 / r)
where v is the speed of m1.
Setting the tension equal to the gravitational force, we can solve for the speed v:
m2 * g = m1 * (v^2 / r)
Now, rearranging the equation to solve for v:
v^2 = (m2 * g * r) / m1
v = sqrt((m2 * g * r) / m1)
Therefore, the required speed at which m1 must rotate in a circle of radius r for m2 to remain hanging at rest is the square root of ((m2 * g * r) / m1).