MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.01
W15D1-1 Bead, Spring, and Hoop Solutions
A bead of mass m slides without
friction on a vertical hoop of radius R .
The bead moves under the combined
action of gravity and a spring, with
spring constant k , attached to the
bottom of the hoop. For simplicity,
assume that the equilibrium length of the
spring is zero. The bead is released from
rest at ! = 0 with a non-zero but
negligible speed to the right. Express
your answer to the following questions
in terms of some or all of the parameters
m , R , k , and the acceleration of gravity
g.
a) What is the speed v of the bead when ! = 90! ?
Solution: Let the zero of gravitational potential energy be at the top of the ring.
b) What is the magnitude and direction of the force the ring exerts on the bead when
! = 90! ?
Solution: N is directed inward, horizontally toward the center of the ring.