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Problems on gravitation, rotational and oscillatory motion

5. Zero, a hypothetical planet, has a mass of 4.0 x 10^23 kg, a radius of 3.0 x 10^6 m, and no atmosphere. A 15 kg space probe is to be launched vertically from its surface.

(a) If the probe is launched with an initial kinetic energy of 5.0 x 10^7 J, what will be its kinetic energy when it is 4.0 x 10^6 m from the center of Zero?

(b) If the probe is to achieve a maximum distance of 8.0 x 10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

7. In 1993 the spacecraft Galileo sent home an image of asteroid 243 Ida and an orbiting tiny moon (now known as Dactyl), the first confirmed example of an asteroid moon system. Other such systems have since been discovered. Assume an asteroid's moon is 1.4 km wide, and that its center is 114 km from the center of the asteroid, which is 55 km long. The moon's orbit is circular with a period of 21 h.

(a) What is the mass of the asteroid?

(b) The volume of the asteroid is 14,100 km^3. What is the density of the asteroid?

10. An object lying on Earth's equator is accelerated in the following three directions.
(a) toward the center of Earth because Earth rotates
(b) toward the Sun because Earth revolves around the Sun in an almost circular orbit
(c) toward the center of our galaxy because the Sun moves about the galactic center

For the latter, the period is 2.5 x 10^8 y and the radius is 2.2 x 10^20 m. Calculate these three accelerations as multiples of g = 9.8 m/s2.

_____ g toward the center of Earth
______ g toward the center of the Sun
_______ g toward the center of the galaxy

11. Some believe that the positions of the planets at the time of birth influence the newborn. Others deride this belief and claim that the gravitational force exerted on a baby by the obstetrician is greater than that exerted by the planets. To check this claim, calculate the gravitational force exerted on a 2.2 kg baby by each of the following.
(a) a 85 kg obstetrician who is 1 m away and roughly approximated as a point mass

(b) the massive planet Jupiter (m = 2 10^27 kg) at its closest approach to Earth (= 6 10^11 m)

(c) Jupiter at its greatest distance from Earth (= 9 10^11 m)

(d) Is the claim correct?

No, the gravitational force due to Jupiter is larger.
Yes, the gravitational force due to the obstetrician is larger.

Chapter 15

1.An object undergoing simple harmonic motion takes 0.35 s to travel from one point of zero velocity to the next such point. The distance between those points is 35 cm.
(a) Calculate the period of the motion.

(b) Calculate the frequency of the motion.

(c) Calculate the amplitude of the motion.

2. An automobile can be considered to be mounted on four identical springs as far as vertical oscillations are concerned. The springs of a certain car are adjusted so that the oscillations have a frequency of 3 Hz.

(a) What is the spring constant of each spring if the mass of the car is 1550 kg and the weight is evenly distributed over the springs?

(b) What will be the vibration frequency if five passengers, averaging 77.0 kg each, ride in the car with an even distribution of mass?

(a) Calculate the torque required to accelerate the Earth in 8 days from rest to its present angular speed about its axis.

(b) Calculate the energy required.

(c) Calculate the average power required.

5. For a 90 kg person standing at the equator, what is the magnitude of the angular momentum about the Earth's center due to Earth's rotation?

6. A girl of mass M stands on the rim of a frictionless merry-go-round of radius R and rotational inertia I that is not moving. She throws a rock of mass m horizontally in a direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is v.

(a) Afterward, what is the angular speed of the merry-go-round?

(b) Afterward, what is the linear speed of the girl?

4. An oscillating block-spring system has a mechanical energy of 1.00 J, an amplitude of 10.4 cm, and a maximum speed of 1.01 m/s.

(a) Find the spring constant.

(b) Find the mass of the block.

(c) Find the frequency of oscillation.

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