Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

Homework Problems

A little old lady and a pro football player collide head-on. Compare their forces on each other, and compare their accelerations. Explain.
The earth is attracted to an object with a force equal and opposite to the force of the earth on the object. If this is true, why is it that when you drop an object, the earth does not have an acceleration equal and opposite to that of the object?
When you stand still, there are two forces acting on you, the force of gravity (your weight) and the normal force of the floor pushing up on your feet. Are these forces equal and opposite? Does Newton's third law relate them to each other? Explain.

In problems 4-8, analyze the forces using a table in the format shown in section 5.3. Analyze the forces in which the italicized object participates.

A magnet is stuck underneath a parked car. (See instructions above.)
Analyze two examples of objects at rest relative to the earth that are being kept from falling by forces other than the normal force. Do not use objects in outer space, and do not duplicate problem 4 or 8. (See instructions above.)

A person is rowing a boat, with her feet braced. She is doing the part of the stroke that propels the boat, with the ends of the oars in the water (not the part where the oars are out of the water). (See instructions above.)

A farmer is in a stall with a cow when the cow decides to press him against the wall, pinning him with his feet off the ground. Analyze the forces in which the farmer participates. (See instructions above.)
A propeller plane is cruising east at constant speed and altitude. (See instructions above.)

Today's tallest buildings are really not that much taller than the tallest buildings of the 1940s. One big problem with making an even taller skyscraper is that every elevator needs its own shaft running the whole height of the building. So many elevators are needed to serve the building's thousands of occupants that the elevator shafts start taking up too much of the space within the building. An alternative is to have elevators that can move both horizontally and vertically: with such a design, many elevator cars can share a few shafts, and they don't get in each other's way too much because they can detour around each other. In this design, it becomes impossible to hang the cars from cables, so they would instead have to ride on rails which they grab onto with wheels. Friction would keep them from slipping. The figure shows such a frictional elevator in its vertical travel mode. (The wheels on the bottom are for when it needs to switch to horizontal motion.)

(a) If the coefficient of static friction between rubber and steel is s, and the maximum mass of the car plus its passengers is M, how much force must there be pressing each wheel against the rail in order to keep the car from slipping? (Assume the car is not accelerating.)

(b) Show that your result has physically reasonable behavior with respect to s. In other words, if there was less friction, would the wheels need to be pressed more firmly or less firmly? Does your equation behave that way?


Unequal masses M and m are suspended from a pulley as shown in the figure.

(a) Analyze the forces in which mass m participates, using a table the format shown in section 5.3. [The forces in which the other mass participates will of course be similar, but not numerically the same.]

(b) Find the magnitude of the accelerations of the two masses. [Hints: (1) Pick a coordinate system, and use positive and negative signs consistently to indicate the directions of the forces and accelerations. (2) The two accelerations of the two masses have to be equal in magnitude but of opposite signs, since one side eats up rope at the same rate at which the other side pays it out. (3) You need to apply Newton's second law twice, once to each mass, and then solve the two equations for the unknowns: the acceleration, a, and the tension in the rope, T.]

(c) Many people expect that in the special case of M = m, the two masses will naturally settle down to an equilibrium position side by side. Based on your answer from part b, is this correct?

(d) Find the tension in the rope, T.

(e) Interpret your equation from part d in the special case where one of the masses is zero. Here "interpret" means to figure out what happens mathematically, figure out what should happen physically, and connect the two.

11 A tugboat of mass m pulls a ship of mass M, accelerating it. The speeds are low enough that you can ignore fluid friction acting on their hulls, although there will of course need to be fluid friction acting on the tug's propellers.

(a) Analyze the forces in which the tugboat participates, using a table in the format shown in section 5.3. Don't worry about vertical forces.

(b) Do the same for the ship.

(c) Assume now that water friction on the two vessels' hulls is negligible. If the force acting on the tug's propeller is F, what is the tension, T, in the cable connecting the two ships? [Hint: Write down two equations, one for Newton's second law applied to each object. Solve these for the two unknowns T and a.]

(d) Interpret your answer in the special cases of M = 0 and M = ∞.

12 Explain why it wouldn't make sense to have kinetic friction be stronger than static friction.

In the system shown in the figure, the pulleys on the left and right are fixed, but the pulley in the center can move to the left or right. The two masses are identical. Show that the mass on the left will have an upward acceleration equal to g/5. Assume all the ropes and pulleys are massless and rictionless.


The figure shows two different ways of combining a pair of identical springs, each with spring constant k. We refer to the top setup as parallel, and the bottom one as a series arrangement.

(a) For the parallel arrangement, analyze the forces acting on the connector piece on the left, and then use this analysis to determine the equivalent spring constant of the whole setup. Explain whether the combined spring constant should be interpreted as being stiffer or less stiff.

(b) For the series arrangement, analyze the forces acting on each spring and figure out the same things.

Solution, p. 279
15 Generalize the results of problem 14 to the case where the two spring constants are unequal.
16 (a) Using the solution of problem 14, which is given in the back of the book, predict how the spring constant of a fiber will depend on its length and cross-sectional area.

(b) The constant of proportionality is called the Young's modulus, E, and typical values of the Young's modulus are about 1010 to 1011. What units would the Young's modulus have in the SI (meterkilogram- second) system?

Solution, p. 280

This problem depends on the results of problems 14 and 16, whose solutions are in the back of the book When atoms form chemical bonds, it makes sense to talk about the spring constant of the bond as a measure of how "stiff" it is. Of course, there aren't really little springs - this is just a mechanical model. The purpose of this problem is to estimate the spring constant, k, for a single bond in a typical piece of solid matter. Suppose we have a fiber, like a hair or a piece of fishing line, and imagine for simplicity that it is made of atoms of a single element stacked in a cubical manner, as shown in the figure, with a center-to-center spacing b. A typical value for b would be about 10-10 m.

(a) Find an equation for k in terms of b, and in terms of the Young's modulus, E, defined in problem 16 and its solution.

(b) Estimate k using the numerical data given in problem 16.

(c) Suppose you could grab one of the atoms in a diatomic molecule like H2 or O2, and let the other atom hang vertically below it. Does the bond stretch by any appreciable fraction due to gravity?

18 In each case, identify the force that causes the acceleration, and give its Newton's-third-law partner. Describe the effect of the partner force. (a) A swimmer speeds up. (b) A golfer hits the ball off of the tee. (c) An archer fires an arrow. (d) A locomotive slows down.
Solution, p. 280

Ginny has a plan. She is going to ride her sled while her dog Foo pulls her. However, Ginny hasn't taken physics, so there may be a problem: she may slide right off the sled when Foo starts pulling.

(a) Analyze all the forces in which Ginny participates, making a table as in section 5.3.

(b) Analyze all the forces in which the sled participates.

(c) The sled has mass m, and Ginny has mass M. The coefficient of static friction between the sled and the snow is 1, and 2 is the corresponding quantity for static friction between the sled and her snow pants. Ginny must have a certain minimum mass so that she will not slip off the sled. Find this in terms of the other three variables.

(d) Under what conditions will there be no solution for M?

20 Example 2 on page 154 involves a person pushing a box up a hill. The incorrect answer describes three forces. For each of these three forces, give the force that it is related to by Newton's third law, and state the type of force.
Solution, p. 280
21 Example 5 on page 170 describes a force-doubling setup involving a pulley. Make up a more complicated arrangement, using more than one pulley, that would multiply the force by a factor greater than two.
22 Pick up a heavy object such as a backpack or a chair, and stand on a bathroom scale. Shake the object up and down. What do you observe? Interpret your observations in terms of Newton's third law.
23 A cop investigating the scene of an accident measures the length L of a car's skid marks in order to find out its speed v at the beginning of the skid. Express v in terms of L and any other relevant variables.
24 The following reasoning leads to an apparent paradox; explain what's wrong with the logic. A baseball player hits a ball. The ball and the bat spend a fraction of a second in contact. During that time they're moving together, so their accelerations must be equal. 176 Chapter 5 Analysis of Forces Newton's third law says that their forces on each other are also equal. But a = F/m, so how can this be, since their masses are unequal? (Note that the paradox isn't resolved by considering the force of the batter's hands on the bat. Not only is this force very small compared to the ball-bat force, but the batter could have just thrown the bat at the ball.)
25 Driving down a hill inclined at an angle θ with respect to horizontal, you slam on the brakes to keep from hitting a deer.

(a) Analyze the forces. (Ignore rolling resistance and air friction.)

(b) Find the car's maximum possible deceleration, a (expressed as a positive number), in terms of g, θ, and the relevant coefficient of friction.

(c) Explain physically why the car's mass has no effect on your answer.

(d) Discuss the mathematical behavior and physical interpretation of your result for negative values of θ.

(e) Do the same for very large positive values of θ.

26 (a) Compare the mass of a one-liter water bottle on earth, on the moon, and in interstellar space. (b) Do the same for its weight.Solution, p. 280

Last Update: 2010-11-11