# Physics

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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. Assume that the equilibrium (relaxed) length of the spring is R. The bead is released from rest at θ = 0 with a non-zero but negligible speed to the right.

(a) What is the speed v of the bead when θ = 90∘ ? Express your answer in terms of m, R, k, and g.

(b) What is the magnitude of the force the hoop exerts on the bead when θ = 90∘ ? Express your answer in terms of m, R, k, and g.

• Physics -

Consider a double star system under the influence of the gravitational force between the stars. Star 1 has mass m1 = 2.42 × 1031 kg and Star 2 has mass m2 = 2.32 × 1031 kg. Assume that each star undergoes uniform circular motion about the center of mass of the system (cm). In the figure below r1 is the distance between Star 1 and cm, and r2 is the distance between Star 2 and cm.

1)If the stars are always a fixed distance s=r1+r2 = 3.12 × 1018 m apart, what is the period of the orbit (in s)?