The piece of exercise equipment shown below includes two handles that are on each end of two parallel springs, each with a stiffness 500 N/m. To build strength, a person holds the two handles and pulls them apart with equal magnitude forces applied in opposite directions to the handles, thus stretching the springs. Neglect the mass of the springs and assume that they are pulled horizontally.

Figure: Two parallel springs between two handles that can be pulled apart.

1. If a person stretches the springs 5 cm from their relaxed lengths and holds it at rest, what is the net force applied to the system and what magnitude force must he apply to each handle?
2. What magnitude force must he apply to each handle to stretch it twice as far (10 cm) and hold it at rest?
3. If the two parallel springs were replaced with a single spring such that the same force would be required to stretch it 10 cm as for the two parallel springs, what would be the stiffness of the single spring?

In the lab, you connect one end of a 15 N/m spring to a ringstand, and you hang a 0.25-kg block from the other end of the spring. It hangs vertically in equilibrium.

1. How far is the spring stretched from its relaxed length?
2. Suppose that you put your hand beneath the block and lift up on the block in order to partially support it. It is still in equilibrium with the spring stretched 5 cm from its relaxed length. What is the force of your hand on the block?

A spring-loaded dart gun is used to shoot a 5-gram dart straight upward. Its spring has a stiffness 50 N/m, and its relaxed length is 5.0 cm. The spring is compressed 4.0 cm from its relaxed length.

1. What is the net force on the dart at the instant the spring is released?
2. What is the net force on the dart at the instant the dart leaves the spring? At this instant, the spring has returned to its relaxed length.

The International Space Station orbits Earth at an altitude of about 350 km (from Earth's surface).

1. What is the magnitude of Earth's gravitational field g at the altitude of the space station?
2. What is the weight of a 90-kg astronaut who is in the space station? (Note weight in this case is defined as the gravitational force by Earth on the astronaut.)

A spring-loaded dart gun is used to shoot a 5-gram dart straight upward. Its spring has a stiffness 50 N/m, and its relaxed length is 5.0 cm. The spring is compressed 4.0 cm from its relaxed length. What is the velocity of the dart when it leaves the spring (i.e. at the instant the spring returns to its relaxed length)? Apply the Momentum Principle iteratively in order to answer the question.

A comet of mass orbits Sun which has a mass . Use the following constants and initial conditions for the comet in your calculations.

Assume that Sun remains stationary during the orbit (i.e. it does not "wobble").

1. What is the position and momentum of the comet at t = 30 days.
2. What is the period of the comet's orbit?
3. What is the position of the comet at aphelion (the point in its orbit when it is furthest from Sun.

A planet of mass kg is at a location m. A star of mass kg is at a location m. What is the gravitational force by the star on the planet?