4.6Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
vCalc Companion Formulas | ||
vCalc Formulary | 4.6 Summary | |
`a_(cm)=F`(total)/m | Acceleration |
weight — the force of gravity on an object, equal to `mg`
inertial frame — a frame of reference that is not accelerating, one in which Newton's first law is true
noninertial frame — an accelerating frame of reference, in which Newton's first law is violated
`F_W` — weight
net force — another way of saying “total force”
Newton's first law of motion states that if all the forces acting on an object cancel each other out, then the object continues in the same state of motion. This is essentially a more refined version of Galileo's principle of inertia, which did not refer to a numerical scale of force.
Newton's second law of motion allows the prediction of an object's acceleration given its mass and the total force on it, `a_(cm)=F`(total)/m. This is only the one-dimensional version of the law; the full-three dimensional treatment will come in chapter 8, Vectors. Without the vector techniques, we can still say that the situation remains unchanged by including an additional set of vectors that cancel among themselves, even if they are not in the direction of motion.
Newton's laws of motion are only true in frames of reference that are not accelerating, known as inertial frames.
j / Problem 6.
1. An object is observed to be moving at constant speed in a certain direction. Can you conclude that no forces are acting on it? Explain. [Based on a problem by Serway and Faughn.]
2. At low speeds, every car's acceleration is limited by traction, not by the engine's power. Suppose that at low speeds, a certain car is normally capable of an acceleration of `3 m"/"s^2`. If it is towing a trailer with half as much mass as the car itself, what acceleration can it achieve? [Based on a problem from PSSC Physics.]
3. (a) Let `T` be the maximum tension that an elevator's cable can withstand without breaking, i.e., the maximum force it can exert. If the motor is programmed to give the car an acceleration `a ( a>0` is upward), what is the maximum mass that the car can have, including passengers, if the cable is not to break?(answer check available at lightandmatter.com)
(b) Interpret the equation you derived in the special cases of `a=0` and of a downward acceleration of magnitude `g`. (“Interpret” means to analyze the behavior of the equation, and connect that to reality, as in the self-check on page 139.)
k / Problem 10, part c.
4. A helicopter of mass `m` is taking off vertically. The only forces acting on it are the earth's gravitational force and the force, `F_"air"`, of the air pushing up on the propeller blades.
(a) If the helicopter lifts off at `t=0`, what is its vertical speed at time`t`?
(b) Check that the units of your answer to part a make sense.
(c) Discuss how your answer to part a depends on all three variables, and show that it makes sense. That is, for each variable, discuss what would happen to the result if you changed it while keeping the other two variables constant. Would a bigger value give a smaller result, or a bigger result? Once you've figured out this mathematical relationship, show that it makes sensephysically.
(d) Plug numbers into your equation from part a, using `m=2300 kg`, `F_"air"=27000 N`, and `t=4.0 s`. (answer check available at lightandmatter.com)
l / Problem 13. The rear wings of the plane collapse under the stress of the catapult launch.
5. In the 1964 Olympics in Tokyo, the best men's high jump was 2.18 m. Four years later in Mexico City, the gold medal in the same event was for a jump of 2.24 m. Because of Mexico City's altitude (2400 m), the acceleration of gravity there is lower than that in Tokyo by about `0.01 m"/"s^2`. Suppose a high-jumper has a mass of 72 kg.
(a) Compare his mass and weight in the two locations.
(b) Assume that he is able to jump with the same initial vertical velocity in both locations, and that all other conditions are the same except for gravity. How much higher should he be able to jump in Mexico City?(answer check available at lightandmatter.com)
(Actually, the reason for the big change between '64 and '68 was the introduction of the “Fosbury flop.”)
6. A blimp is initially at rest, hovering, when at `t=0` the pilot turns on the engine driving the propeller. The engine cannot instantly get the propeller going, but the propeller speeds up steadily. The steadily increasing force between the air and the propeller is given by the equation `F=kt`, where `k` is a constant. If the mass of the blimp is `m`, find its position as a function of time. (Assume that during the period of time you're dealing with, the blimp is not yet moving fast enough to cause a significant backward force due to air resistance.)(answer check available at lightandmatter.com) ?
m / Problem 15.
7. (solution in the pdf version of the book) A car is accelerating forward along a straight road. If the force of the road on the car's wheels, pushing it forward, is a constant 3.0 kN, and the car's mass is 1000 kg, then how long will the car take to go from 20 m/s to 50 m/s?
8. Some garden shears are like a pair of scissors: one sharp blade slices past another. In the “anvil” type, however, a sharp blade presses against a flat one rather than going past it. A gardening book says that for people who are not very physically strong, the anvil type can make it easier to cut tough branches, because it concentrates the force on one side. Evaluate this claim based on Newton's laws. [Hint: Consider the forces acting on the branch, and the motion of the branch.]
9. A uranium atom deep in the earth spits out an alpha particle. An alpha particle is a fragment of an atom. This alpha particle has initial speed `v`, and travels a distance `d` before stopping in the earth.
(a) Find the force, `F`, from the dirt that stopped the particle, in terms of `v,d`, and its mass, `m`. Don't plug in any numbers yet. Assume that the force was constant.(answer check available at lightandmatter.com)
(b) Show that your answer has the right units.
(c) Discuss how your answer to part a depends on all three variables, and show that it makes sense. That is, for each variable, discuss what would happen to the result if you changed it while keeping the other two variables constant. Would a bigger value give a smaller result, or a bigger result? Once you've figured out this mathematical relationship, show that it makes sensephysically.
(d) Evaluate your result for `m=6.7×10^(?27) kg`, `v=2.0×10^4 km"/"s`, and `d=0.71 mm`.(answer check available at lightandmatter.com)
10. You are given a large sealed box, and are not allowed to open it. Which of the following experiments measure its mass, and which measure its weight? [Hint: Which experiments would give different results on the moon?]
(a) Put it on a frozen lake, throw a rock at it, and see how fast it scoots away after being hit.
(b) Drop it from a third-floor balcony, and measure how loud the sound is when it hits the ground.
(c) As shown in the figure, connect it with a spring to the wall, and watch it vibrate.
(solution in the pdf version of the book)
11. While escaping from the palace of the evil Martian emperor, Sally Spacehound jumps from a tower of height `h` down to the ground. Ordinarily the fall would be fatal, but she fires her blaster rifle straight down, producing an upward force of magnitude `F_B`. This force is insufficient to levitate her, but it does cancel out some of the force of gravity. During the time `t` that she is falling, Sally is unfortunately exposed to fire from the emperor's minions, and can't dodge their shots. Let `m` be her mass, and `g` the strength of gravity on Mars.
(a) Find the time `t` in terms of the other variables.
(b) Check the units of your answer to part a.
(c) For sufficiently large values of `F_B`, your answer to part a becomes nonsense --- explain what's going on.(answer check available at lightandmatter.com)
12. When I cook rice, some of the dry grains always stick to the measuring cup. To get them out, I turn the measuring cup upside-down and hit the “roof” with my hand so that the grains come off of the “ceiling.” (a) Explain why static friction is irrelevant here. (b) Explain why gravity is negligible. (c) Explain why hitting the cup works, and why its success depends on hitting the cup hard enough.
13. At the turn of the 20th century, Samuel Langley engaged in a bitter rivalry with the Wright brothers to develop human flight. Langley's design used a catapult for launching. For safety, the catapult was built on the roof of a houseboat, so that any crash would be into the water. This design required reaching cruising speed within a fixed, short distance, so large accelerations were required, and the forces frequently damaged the craft, causing dangerous and embarrassing accidents. Langley achieved several uncrewed, unguided flights, but never succeeded with a human pilot. If the force of the catapult is fixed by the structural strength of the plane, and the distance for acceleration by the size of the houseboat, by what factor is the launch velocity reduced when the plane's 340 kg is augmented by the 60 kg mass of a small man?(answer check available at lightandmatter.com)
14. The tires used in Formula 1 race cars can generate traction (i.e., force from the road) that is as much as 1.9 times greater than with the tires typically used in a passenger car. Suppose that we're trying to see how fast a car can cover a fixed distance starting from rest, and traction is the limiting factor. By what factor is this time reduced when switching from ordinary tires to Formula 1 tires?(answer check available at lightandmatter.com)
15. In the figure, the rock climber has finished the climb, and his partner is lowering him back down to the ground at approximately constant speed. The following is a student's analysis of the forces acting on the climber. The arrows give the directions of the forces.
force of the earth’s gravity,? |
force from the partner’s hands,? |
force from the rope,? |
The student says that since the climber is moving down, the sum of the two upward forces must be slightly less than the downward force of gravity.
Correct all mistakes in the above analysis. (solution in the pdf version of the book)
Equipment:
Suppose a person pushes a crate, sliding it across the floor at a certain speed, and then repeats the same thing but at a higher speed. This is essentially the situation you will act out in this exercise. What do you think is different about her force on the crate in the two situations? Discuss this with your group and write down your hypothesis:
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1. First you will measure the amount of friction between the wood block and the butcher paper when the wood and paper surfaces are slipping over each other. The idea is to attach a spring scale to the block and then slide the butcher paper under the block while using the scale to keep the block from moving with it. Depending on the amount of force your spring scale was designed to measure, you may need to put an extra mass on top of the block in order to increase the amount of friction. It is a good idea to use long piece of string to attach the block to the spring scale, since otherwise one tends to pull at an angle instead of directly horizontally.
First measure the amount of friction force when sliding the butcher paper as slowly as possible: --------------------------
Now measure the amount of friction force at a significantly higher speed, say 1 meter per second. (If you try to go too fast, the motion is jerky, and it is impossible to get an accurate reading.) ------------------------------------------------------------------------------
Discuss your results. Why are we justified in assuming that the string's force on the block (i.e., the scale reading) is the same amount as the paper's frictional force on the block?
2. Now try the same thing but with the block moving and the paper standing still. Try two different speeds.
Do your results agree with your original hypothesis? If not, discuss what's going on. How does the block “know” how fast to go?
4.6Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.