### Work and Energy

Hints – See this image from Wikipedia of the earth’s orbit around the sun. Think about how “work” is defined in physics (see the first answer on this page and the references therein).

You can also start by thinking about energy – the total energy will be conserved, but what about the kinetic energy of the earth alone? And how does the behavior of the earth’s kinetic energy relate to the work done on it? (Neglect the effect of the other planets.)

Solution resources – See this post on stack exchange (it has a nice illustration). On this page, there is a brief discussion of elliptical orbits (starting with #7) and work done (# 10).

This short YouTube video applies conservation of energy to elliptical orbits.

This video by Dr. Elliot Schneider derives the orbit of the Earth using conservation of energy and conservation of angular momentum.

Hints – Consider the rolling object as a (rigid) system of many particles. Consider separately, the case where the object is rolling at constant speed and the case where it is accelerating (eg: rolling down an incline). In each case, what are the forces acting on the object? What is the displacement of the component of the system the friction force is acting on?

Answers – This answer on stack exchange discusses the situation with friction for an object rolling with constant velocity.

This video by Dr. Peter Dourmashkin of MIT compares the friction on a wheel both when it is rolling at constant velocity vs accelerating down an incline.

This video by Dr. Peter Dourmashkin of MIT discusses friction and work for an object rolling down an inclined plane (accelerating velocity).

This answer on stack exchange discusses friction and work done for a rolling object accelerated by an external force.

Hints – Draw a diagram showing all forces acting on the system. Think carefully about which “particle” of the system your force is acting on and how much it is being displaced.

This Wikipedia article has a nice discussion of pulley arrangements that provide mechanical advantage.

Answers – This YouTube video discusses pulleys, mechanical advantage, and work done.

This note by Tom Weideman from UC Davis discusses mechanical advantage and work done.

Hints – You can tackle this problem by analyzing forces and using Newton’s 2nd law, though it is simpler to solve by applying the principle of conservation of energy. Are the forces which do work conservative? What is the potential energy? What is the expression for the kinetic energy of each object? Kinetic energy due to rigid rotation is discussed in section 19-4 of Feynman’s lectures and in this Wikipedia article. Remember that an object rolling without slipping also has a kinetic energy associated with the linear motion of its center of mass.

Answers – This entry from hyperphysics discusses the race between a hoop and disc.

This note by Andrew Duffy of Boston University discusses a race between several different objects rolling down a ramp.

This video by Dr. Peter Dourmashkin of MIT uses conservation of energy to obtain the velocity of an object rolling down an inclined plane.

This video by Dr. Peter Dourmashkin of MIT uses derives the kinetic energy of an object which is both translating and rotating.

This video by Dr. Peter Dourmashkin of MIT uses force and torque methods to obtain the velocity of an object rolling down an inclined plane.

### Momentum and Center of Mass

Hints – Think about conservation of momentum. See this answer.

Answers – See this Wikipedia section on perfectly inelastic collisions.

A solution to this problem in the case with a friction force provided on this website.

This short video by Dr. Peter Dourmashkin of MIT analyzes a totally inelastic collision.

Hints – Think about the angular momentum right before and after the collision and use its conservation. Conservation of momentum will allow you to figure out the linear motion. Note that the system will spin about its center of mass (why?).

Remember that the kinetic energy of a rigid body receives contributions both from its linear motion and its rotational motion (see this Wikipedia article).

Solution resources – This YouTube video treats the analogous problem of a bullet hitting a rod lying on ice. The analysis of the case of the bullet hitting a solid or hollow disc is exactly analogous, the only difference being the different moments of inertia involved.

This video by Dr. Peter Dourmashkin of MIT derives the kinetic energy of a body which is both translating and rotating.

Hints – Think about the center of mass. See this excellent discussion on the center of mass in section 18-1 of Feynman’s lectures (notice his example of a man walking on a spaceship).

Solution resources – See this stack exchange post for the solution with a very similar application of the center of mass.

See this YouTube video for an analysis of the analogous “man on a boat” problem.

Hints – Think about the conservation of angular momentum. To simplify the problem imagine that she is holding large masses in her hands while twirling them and that her arms are “massless”. Pick a point and calculate the angular momentum of these masses about it.

Solution resources – Try and figure out the angular momentum vector for the rotation of each hand about say the center of her shoulders. Then add these together to get the total angular momentum. Her body will rotate to compensate since the initial angular momentum was zero.

This lecture note discusses the angular momentum vector in an analogous setup (compare the rotation of their conical pendulum with one of her arms doing circles).

### Oscillation

Hints – Draw a diagram of all the forces and apply Newton’s 2nd law to this system to write down the equation for oscillations about the equilibrium position. What frequency do solutions to this equation have? (See this Wikipedia article.)

Answers – This note by Prof. Martin of Queens University has a clear analysis of oscillations of the vertically hanging spring.

This note by Prof. John Deutsch of UCSC analyzes the oscillation of a vertically hanging spring.

This short video by Prof. Matt Anderson compares horizontal and vertical springs.

Hints – Approximate the Earth as a solid sphere with a uniform mass density. You will need to figure out the gravitational force/potential energy at points inside the sphere (see the beginning of this article on the “shell theorem”; this theorem in turn can be derived quickly using the “Gravitational Gauss law“). Draw a diagram of the forces on an object in the frictionless tunnel and use Newton’s 2nd law to write down an equation for motion along the tunnel. (For the first part, think about the conservation of energy.)

Answers – This Wikipedia article contains a derivation of the answer. Such a tunnel is often called a “gravity train”.

This video by Prof. Walter Levine analyzes a tunnel through the earth.

### Equilibrium (Statics)

Hints – Treat the stick as a collection of discrete particles. Write the displacement of a particle from the center of the stick in terms of its displacement from the end of the stick, in order to relate the torques.

Answers – This note by Tom Weideman of UC Davis relates the torque about two different points for a rigid body.

Hints – Start with a single brick. Suppose you place the center (of mass) of this brick right at the edge of the table – think carefully about which forces act on the brick. Is the total torque on the brick zero? What happens if the center moves over the edge?

Next picture the stack with two bricks and imagine holding the lower brick in place – how far can the upper brick slide over the lower brick before it tips over? Then, consider the whole two brick stack as one system and imagine sliding it back and forth. Where must the center of mass of this system lie relative to the edge of the table so that it does not tip over (the torque on it is zero)? Repeat the reasoning for three bricks, starting with the top one. Can you generalize the pattern to how far a stack of N bricks can extend?

Answers – This entry in an online textbook (Interactive Real Analysis) has a nice discussion of the block stacking problem.

This post on the Data Genetics blog for an analysis of this block stacking problem.

This video by Albert Meyer of MIT analyzes the stacking problem.

This YouTube video by Physics Ninja also analyzes the stacking problem.

### Gravitation

Hints – See the this Wikipedia article on the “shell theorem”. This theorem can be derived quickly from the “Gravitational Gauss law“.

Answers – This section of the Wikipedia article on the Shell Theorem derives the gravitational force outside a shell.

See this YouTube video by DrSmithPhysics for an intuitive explanation of why the gravitational force vanishes inside the shell.

I have taught physics at levels ranging from introductory classical mechanics to advanced graduate quantum mechanics, along with calculus and linear algebra.