Question 1- At point A the wooden block below is traveling at 2.3 m/s. The coefficient of kinetic friction between the level table and the block is 0.22. The block slides and then falls off the table.

a) How much work does friction do between point A and point B (just before it hits the floor)?

b) How much work is done by gravity between points A and B?

c) Use the Work-­-Energy Theorem to determine the kinetic energy of the block at B.

Question 2- A 60 gram bead slides on a bent wire as shown. Point A is 0.56 m higher than point C and friction acts on the bead as it slides. The velocity of the bead is shown on the diagram at points A and C.

a) Draw a free body diagram of the bead at the three points shown (A,B and C). Which of the forces (if any) in the diagram do work on the bead as it slides from A to C? Which forces (if any) do not? Briefly explain answers.

b) How much energy is converted to non-mechanical form (mostly thermal) through friction (work done by) between points A and C?

Question 3- Imagine a metal bead of mass m (with a hole cut through its center) that slides along a wire as shown. If the wire has a parabolic shape then it can be described by a curve in the y-x plane as shown: y = c- bx+ ax2 where a, b and c are all positive constants. The position of the bead at any point on the wire can then be described by the vector r = xˆi + (c- bx+ ax²)ˆj  in unit vector notation. Note that the y component of this vector is just the parabolic function y(x) described above. When the particle undergoes an infinitesimal displacement, as in the diagram, moving from position r₁ to position r₂, its displacement vector is given by: dr = dxˆi + (-b+2ax)dxˆj  (take derivative with respect to x of each component of  r and then multiply by dx).

a) The work done by the force of gravity, F_g = - mgˆj, as the bead slides along the wire from position r₁ to position r₂ is given by _r1∫^r2Fg·dr. Determine the dot product: Fg·dr. (No numerical substitutions please). This is the integrant of the integral (above) that defines the work done by the force of gravity along the parabolic path.

b) If you integrate your result from part a from x coordinate x₁ to x coordinate x₂ you will have calculated the work done by F_g as the bead slides from position r₁ to position r₂ along the wire. Without numerical substitution perform this integration and find the work done by F_g as the bead slides from x₁ to x₂. (You'll get a formula that depends on x₁ and x₂).

c) Let Δy = y(x₂) - y(x₁) with y(x) being the parabolic function described above. Show algebraically that we could get the result from part b from the formula: W = - mgΔy. Which means that the work done by the force of gravity only depends on the vertical displacement?

Question 4- A pendulum consists of a metal sphere, of mass m, attached to a rigid rod that is attached to the ceiling as shown below. A person applies a force F_ext to the ball that varies in both magnitude and direction while the ball rises, following the arc of a circle, at constant speed, through a vertical distance of 0.8 m.

For parts a, b and c think of the "system" as the sphere.

a) What forces act on the sphere? Sketch a free body diagram and carefully label each force. A student in class says that the external force must be equal and opposite to the force of gravity throughout the process described since the sphere moves at constant speed. Do you agree? If not, is there another other force that acts vertically on the sphere? Discuss.

b) Which of the forces from part a do work? Which don't? Explain your reasoning. Does the kinetic energy of the sphere change over the course of the motion described? How do you know?

c) In terms of the mass m (m is a number, but don't make up a value!) how much work (including the correct sign) is done by gravity during the process? How much work is done by the external force applied by the person? Are these quantities related? If so, how? Discuss.

For part d think of the "system" as the sphere and earth's gravity field.

d) In terms of the mass m how much gravitational potential energy is gained by the sphere (+earth's gravity field) during the process? How much chemical energy, minimum was consumed within the person's body? How does the law of conservation of energy apply to this situation?

You need to clearly explain your answers for all parts of this question, calculations alone are not sufficient.