more physics q's i need help with

Question

1. a solar probe is orbiting the sun and its closest approach brings it approximately 1.8 x 10^10 m from the center of the sun. At this point in its orbit, it travels at a speed of 110,000 m/s. what would be its linear velocity when the probe is 3.2 x 10^10 m from the sun's center in the course of its orbit? The answer is 62000 m/s but how do you get to his answer?
2. You hang a 10 kg mass on a spring and the spring stretches 15 cm due to its weight. if you pull the 10 kg mass down a further 5.0 cm and release, what will be the period of the vibration? The answer is 0.78 s; how do you get to this answer?
3. Two wheels of different sizes are in contact at their outer edges and able to cause the other to turn without slipping. If the larger wheel has a radius of 15 cm and an angular velocity of 60 rad/s, and the smaller wheel has a radius of 5 cm, what would the smaller wheel's angular velocity be? The answer is 180 m/s; how do you arrive at this answer?

Any kind of help would be appreciated

Answer 1

(1)

Angular momentum must be conserved, so

mv₁r₁ = mv₂r₂ ⇒

v₂ = v₁r_{1 /} r₂ = (110,000)(1.8 x 10¹⁰) / (3.2 x 10¹⁰) m/s = 61875 m/s (round off to 62000 m/s)

(2)

First, find the spring stiffness.

kx = mg ⇒

k = mg/x = (10)(9.8) / 0.15 N/m = 653 N/m

Now get the period T knowing k.

T = 2π √(m/k) = 2π √(10/653) s = 0.78 s (with roundoff)

Note the additional pull to 5.0 cm does not affect the period.

(3)

They must have the same tangential velocity, so

ω₂r₂ = ω₁r₁ ⇒

ω₂ = ω₁r₁ / r₂ = (60)(0.15) / 0.05 rad/s = 180 rad/s

Note the answer to (3) provided in the question has incorrect units. The units should be rad/s.