A 660-g hoop 95 cm in diameter is rotating at 170 rpm about its central axis. What’s its angular momentum?
- A potter’s wheel with rotational inertia 6.20 kg • m² is spinning freely at 20.0 rpm. The potter drops a 2.50-kg lump of clay onto the wheel, where it sticks 48.0 cm from the rotation axis. What’s the wheel’s subsequent angular speed?
The largest value of mM\frac{m}{M} for which the hoop never rises off the ground is: mM=310\boxed{\frac{m}{M} = \frac{3}{10}}
Explanation
We are dealing with a dynamic system: two identical beads of mass mm, initially at the top of a vertical frictionless hoop of mass MM and radius RR. The hoop is on the ground, and the beads are free to slide along the hoop. When slightly disturbed, they slide symmetrically in opposite directions (one clockwise, one counterclockwise).
Key Idea:
As the beads slide, they exert normal forces on the hoop due to the centripetal acceleration needed to keep them on the circular path. By Newton’s third law, these normal forces act back on the hoop and can generate a vertical component that may cause the hoop to lift off the ground. To ensure the hoop remains in contact, the net upward force from the beads’ normal forces must never exceed the weight of the hoop.
Forces and Motion:
Let θ\theta be the angular position of the beads from the vertical. As the beads slide to angle θ\theta, they must experience centripetal acceleration ac=v2Ra_c = \frac{v^2}{R}. The normal force N(θ)N(\theta) from the hoop provides this. The vertical component of each bead’s normal force is N(θ)cosθN(\theta) \cos \theta, directed upward on the hoop.
There are two beads, so the total upward force on the hoop is 2N(θ)cosθ2N(\theta) \cos \theta. If this exceeds the hoop’s weight MgMg, the hoop lifts.
Critical Point:
At the angle where 2N(θ)cosθ2N(\theta)\cos\theta is maximum, we equate it to MgMg to find the critical mass ratio. Analyzing the energy and dynamics, we find: mM=310\frac{m}{M} = \frac{3}{10}
This is the maximum ratio of bead to hoop mass such that the hoop remains grounded. For larger values, the vertical normal force from the beads can lift the hoop off the surface.
