A spring is attached between the end of a low-friction track and the front of a 1.2 kg cart. A second identical spring is attached to the back of the cart and the other end of the track. When in equilibrium, each spring is stretched 0.4 m from its unstretched position. The cart is displaced 0.15 m from equilibrium and released from rest. It oscillates with a period of 3.0 s. x is the position of the cart at any time t, with x = 0 defined as the equilibrium position of the cart.

What is the angular frequency of the system?

What is the effective spring stiffness of the system?

What is the total energy of the oscillator?

What is the maximum speed of the cart?

When the cart is 0.075 m from its equilibrium position, what percentage of the total energy is elastic potential energy and what percentage is kinetic energy?

1610001 Stiffness of an airbag on a Mars rover 1610001

When the Mars rovers Spirit and Opportunity landed on Mars, they landed via an airbag. Basically, as they were falling, an airbag deployed and they bounced off the surface of Mars until they finally came to rest. So that they didn't have too much kinetic energy upon impact, thrusters were initially fired in order to slow them down before the airbags were deployed.

The airbag can be modeled as a spring because when the rover and airbag hit the surface of mars, the airbag compresses much like a spring would compress.

Suppose that a 250 kg rover is dropped onto the surface of Mars. At a height of 10 m, the thrusters are turned off and it has a downward speed of 5 m/s. When it hits the surface of Mars, the airbag compresses 0.5 m at full impact before the system then rebounds upward. The acceleration due to gravity near the surface of Mars is . What is the "stiffness" of the airbag if you model the airbag as an ideal spring?

16a0002 Projectile motion with air resistance 16a0002

The force of air on a moving object can be quite complicated. Imagine the force of air on a spinning golf ball, for example, which can be hooked, sliced, hit with top-spin or back-spin, etc. However, we are going to use a model for air resistance that assumes that the force of air on a certain ball is

where the coefficient is in units of and speed is in m/s. Note that for low speeds, this force is very small in comparison to the gravitational force on a more massive object. However, for a 0.002-kg nerf ball thrown with an initial speed of 5.55 m/s, the force of air on the ball at the instant it is thrown is 0.0308 N. That is bigger than its weight! Therefore, the force of air on the ball cannot be neglected.

Suppose you toss a 0.002 kg nerf ball to a friend. The ball leaves your hand with a speed of 5.55 m/s at an angle, with respect to the horizontal, of . Define the origin to be the position of the ball at the instant it leaves your hand. Calculate the position and velocity of the ball between t = 0 and t = 0.1 s:

If the ball leaves your hand at a height of 1.5 m from the ground, what is the clock reading when the ball hits the ground and how far from your hand (horizontally) does the ball land?

16a0001 Force by air resistance on a skydiver 16a0001

Suppose that a skydiver falls from an airplane, with her arms and legs outward in a horizontal plane as she falls downward. Her mass is 65 kg and she eventually reaches terminal speed of 55 m/s. Assume that the drag force on her has a magnitude where is a constant that depends on the density of air, the cross-sectional area of the skydiver, and the drag coefficient of the skydiver.

What is the constant for the skydiver?

Suppose that the constant D is approximately the same for another skydiver of mass . What will be his terminal speed?

1640001 Equilibrium temperature of a cup of coffee 1640001

If 0.5 kg of coffee at is poured into a 1-kg glass mug ( ) at , what will be the equilibrium temperature of the system if the system is thermally insulated from its surroundings?

1920001 A "rear-end" collision of two carts on a track. 1920001

In the lab, a 1.0 kg cart traveling in the +x direction with a speed of 0.5 m/s collides with a 2.0 kg cart that is moving in the +x direction at 0.2 m/s. If the velocity of the more massive cart after the collision is 0.3 m/s, what is the change in internal energy of the system of the two carts?

1820001 Elastic collision of an alpha particle and an oxygen nucleus. 1820001

A car of mass 500 kg and a truck of mass 1200 kg are moving directly toward each other at the same speed of 20 m/s. Define the +x direction to the direction of the car.

What is the velocity of the center of mass of the system, defined to be the car and truck?

At the moment when the vehicles are 100 m apart, what is the location of the center of mass of the system? Treat the vehicles as point particles.

1740001 Energy and amplitude of vibration of HCl 1740001

HCl is a diatomic molecule that can be modeled as two balls connected by a spring, i.e. a quantum oscillator. Experiments indicate that the spring constant k of the bond is 480 N/m. Assume the reduced mass of the molecule to be about . Treat the molecule as a quantum oscillator of stiffness and mass .

Determine the energies of the two lowest-energy vibrational states, in units of eV. (That's the ground state and first excited state.)

For each energy state, determine the amplitude of vibration of the oscillator if you treat it classically with an energy .

What energy photon, in units of eV, is emitted if the molecule transitions from the first excited vibrational state to the ground state?

What is the change in the amplitude of vibration if the molecule transitions from the first excited vibrational state to the ground state?

After the molecule emits the photon, it loses energy. Does its frequency of oscillation increase, decrease, or remain constant? Explain your answer.

1710001 Infrared photons absorbed by hydrogen 1710001

Suppose that you shine continuous IR light in the range 0.5 eV - 1.6 eV on atomic hydrogen gas and study the light that gets absorbed by the gas. What are the two lowest energy photons that will be absorbed by atoms and what are the initial and final states of the hydrogen atoms that absorb these photons?