Physics 1: Work and Energy

The 100 kg box shown below is being pulled along the x-axis by a student. The box slides across a rough surface, and its position $x$ varies with time $t$ according to the equation $x = 0.5t^3 + 2t$ , where $x$ is in meters and $t$ is in seconds.

C. Calculate the net work done on the box in the interval $t$ = 0 to $t$ = 2 s would be greater than, less than, or equal to the answer in part (C). Justify your answer.

The figure below depicts a roller coaster. Assume the roller coaster starts at a velocity of 0 m/s from 70 m.

B. Determine the height of the track at point x. It is known that the roller coaster has a velocity of 30 m/s at x.

A rubber ball of mass $m$ is dropped from a cliff. As the ball falls, it is subject to air drag (a resistive force caused by the air). The drag force on the ball has a magnitude $bv^2$ , where $b$ is a constant drag coefficient and $v$ is the instantaneous speed of the ball. The drag coefficient $b$ is directly proportional to the cross-sectional area of the ball and the density of the air and does not depend on the mass of the ball. As the ball falls, its speed approaches a constant value called the terminal speed.

A. Draw and label all the forces on the ball at some instant before it reaches terminal speed.

B. State whether the magnitude of the acceleration of the ball of mass $m$ increases, decreases, or remains the same as the ball approaches terminal speed. Explain.

C. Write, but do NOT solve, a differential equation for the instantaneous speed $v$ of the ball in terms of time $t$ , the given quantities, and fundamental constants.

D. Determine the terminal speed $v_t$ in terms of the given quantities and fundamental constants.

E. Determine the energy dissipated by the drag force during the fall if the ball is released at height $h$ and reaches its terminal speed before hitting the ground, in terms of the given quantities and fundamental constants.

In the system of two blocks and a spring shown below, blocks 1 and 2 are connected by a string that passes over a pulley. The initially unstretched spring connects block 1 to a ridged wall. Block 1 is released from rest, initially slides to the right, and is eventually brought to rest by the spring and the friction on the horizontal surface. Which of the following is true of the energy of the system during the process?

E. The potential energy lost by block 2 is greater in magnitude than the potential energy gained by the spring

A rope of length $L$ is attached to a support at point C. A person of mass $m_1$ sits on a ledge at position A holding the other end of the rope so that it is horizontal and taut, as shown below. The person then drops off the ledge and swings down on the rope toward position B on a lower ledge where an object of mass $m_2$ is at rest. At position B the person grabs hold of the object and simultaneously lets go of the rope. The person and object then land together in the lake at point D, which is a vertical distance $L$ below position B. Air resistance and the mass of the rope are negligible. Derive expression for each of the following in terms of $m_1$ , $m_2$ , $L$ , and $g$.

E. The total horizontal displacement x of the person from position A until the person and object land in the water at point D.

A 50 kg object is to be launched 7 x 10$^6$ m out into space (not into orbit, it will fall back to earth). Given that the radius of the earth is 6.4 x 10 $^6$ m and the mass of the earth is 6.0 x 10 $^{24}$ kg, find:

A. The work done to put the object that far into space.

B. The total kinetic energy of the object required to put the object that far into space.

C. The binding energy.

D. The escape velocity.

A 60 kg object is to be launched into orbit, 7 x 10$^6$ m above the Earth's surface. Given that the radius of the earth is 6.4 x 10 $^6$ m and the mass of the earth is 6.0 x 10 $^{24}$ kg, find:

A. The work done to put the object into orbit.

B. The velocity required to put the object into orbit.

C. The velocity of the object once it is in orbit.