A mass m is attached to a cord passing through a small hole in a frictionless, horizontal surface. The mass is initially orbiting with speed vi in a circle of radius ri. The cord is then slowly pulled from below, and the radius of the circle decreases to r. (Use r_i for ri, v_i for vi, m, and r as appropriate in your equations below.)
(a) What is the speed of the mass when the radius is r?
(b) Find the tension in the cord as a function of r
(c) How much work W is done in moving m from ri to r? (Note: The tension depends on r.)
Do you just use conservation of energy for this? Is it just rotational energy?
use conservation of angular momentum, it is easier. Angularmomentum=Vi/ri * mi
But you can use energy if you wish.
Yes, for this problem, you can use the conservation of energy to find the answers. Since the mass is moving in a circle, we can consider its rotational energy.
(a) To find the speed of the mass when the radius is r, we can equate the initial and final kinetic energies. The initial rotational kinetic energy is given by:
KEi = (1/2) * m * v_i^2
The final rotational kinetic energy is:
KEf = (1/2) * m * v_f^2
Since the mass is attached to a cord passing through a hole, the tension in the cord always provides the centripetal force keeping the mass in circular motion. The centripetal force is given by:
F = m * a = m * v_f^2 / r
where a is the centripetal acceleration and v_f is the final speed. Rearranging this equation, we can solve for v_f:
v_f^2 = F * r / m
Substituting this expression for v_f in the equation for KEf, we get:
KEf = (1/2) * m * (F * r / m) = (1/2) * F * r
Equating KEi and KEf, we have:
(1/2) * m * v_i^2 = (1/2) * F * r
Solving for v_f, we find:
v_f = v_i * (ri / r)
So, the speed of the mass when the radius is r is given by v_f = v_i * (ri / r).
(b) To find the tension in the cord as a function of r, we can consider the forces acting on the mass. At any point, the tension T in the cord provides the centripetal force. Thus, we have:
T = m * v_f^2 / r = m * (v_i * (ri / r))^2 / r = m * v_i^2 * (ri^2 / r^3)
So, the tension in the cord is T = m * v_i^2 * (ri^2 / r^3).
(c) To find the work done in moving the mass from ri to r, we can use the definition of work as the change in kinetic energy. The work done is equal to the final kinetic energy minus the initial kinetic energy:
W = KEf - KEi = (1/2) * m * v_f^2 - (1/2) * m * v_i^2
Substituting the expression for v_f, we get:
W = (1/2) * m * (v_i * (ri / r))^2 - (1/2) * m * v_i^2
Simplifying this expression, we find:
W = (1/2) * m * v_i^2 * ((ri / r)^2 - 1)