Practice Problems

1. Consider a 0.1-kg coconut, falling from a tree 4-m above the ground. Find the following:

a. type (and amount) of energy before it falls
b. type (and amount) of energy right before it hits the ground
c. velocity right before it hits the ground
d. velocity at a point 1-m above the ground

2. Two toy cars collide on a track. A red car (0.2-kg) is traveling at 4 m/s and it smacks into the back end of a blue car (0.4 kg) traveling at 2 m/s in the same direction. They stick together. Find:

a. the type of collision that this is
b. the velocity of the red/blue cars (stuck together) immediately after the collision
c. what happens to the kinetic energy of the system before and after the collision. Calculate it, if you're not sure

3. Explain the following:

a. conservation of energy
b. conservation of momentum
c. elastic vs. inelastic collisions
d. the difference between mass and weight (and how to calculate it)

4. Two cars spring (explode) apart. If the first (1-kg) moves to the left at a final speed of 3 m/s, what is the speed of the second car (0.5-kg)?

5. Revisit the Newton's 2nd Law track problem: A 0.5-kg cart is pulled across a track by a string. The string is connected over a pulley to a 0.1-kg mass.

a. Find the acceleration of the cars
b. Find the acceleration of the cars if there is 0.5-N of friction between the car and track

6. How fast would a rollercoaster car have to travel so that you would just lose contact with the seat over a 5-m (radius) hill?

Session 10 - Energy

Session 9 - Newton's Second Law

As we have seen, Newton's second law is easy to express mathematically:

F = m a

Strictly speaking, the force (F) is the NET FORCE. The mass (m) refers to the mass of the system, and the acceleration (a) also refers to the system.

The unit is the kg m / s^2

This is defined as a newton (N).

We consider a few cases in detail.

1. Weight

First, the force due to gravity - weight, W

Since F = m a, and the acceleration is due to gravity,

W = m g

Mass (m) is the amount of matter (or stuff). The weight is a way of quantifying how much gravity pulls on the mass. This means, if g is different, the weight is also different. You weigh less on the Moon, more on Jupiter, less at high altitudes, etc.

2. Inclined Planes

Objects resting on inclined planes are not free to fall directly down. Rather, they are constrained by the geometry of the plane. Part of the weight (a parallel component) can be thought of as acting DOWN the plane. Part of the weight (a perpendicular component) can be thought of as acting ONTO the plane. By trig:

W(parallel) = mg sin(theta)

W(perpendicular) = mg cos(theta)

Without any resistance whatsoever, all objects slide down a plane with the same acceleration:

F = m a

W(parallel) = m a

mg sin(theta) = m a

a = g sin(theta)

3. Friction

Friction is a catch-all term for any resistive force (really due to electromagnetic interactions between surfaces).

The frictional force (f) is the amount of force that resists motion - that is, it acts in the direction opposite the motion. We can quantify friction by introducing a coefficient of friction (u).

u = f / mg

Meaning: the ratio of frictional force that exists to the weight of the object is defined as the coefficient of friction. Typically, this is a (unitless) number much less than 1.

4. Circular Motion

As Newton's 1st law would predict, any object that moves in a circular path has a REASON to do that - some force is causing it to happen. Recall that acceleration is a change in velocity - since velocity refers to a magnitude (speed) AND a direction, if the direction of a body is changing, it MUST be accelerating even if the speed does not change.

Consider a ball spinning on a string at a constant speed. Even if the speed remains constant, we know that the ball is accelerating - its direction is constantly changing. Some force must be causing that to happen. We call such a force - a center-directed force - centripetal. The center-directed acceleration that results is called centripetal acceleration (ac).

ac = v^2 / r

Or, to compute the magnitude of the centripetal acceleration, we take the speed squared and divide it by the radius of orbit.

The units of acceleration are still m/s^2.

Test practice problems, without answers

1. How high will a ball travel into the air, if thrown straight up at 25 m/s?

2. If this ball were thrown at a 40-degree angle (also at 25 m/s), find the following: time in air, horizontal displacement, max vertical displacement

3. Add the following vectors together, which are acting concurrently at a right angle to each other: 40 m/s and 60 m/s. Find the angle from the 40 m/s vector.

Session 7 part 3 - some historical info FYI

http://en.wikipedia.org/wiki/Isaac_Newton

This is really exhaustive - only for the truly interested.

This one is a bit easier to digest:

http://galileoandeinstein.physics.virginia.edu/lectures/newton.html

We'll return to Newton's gravitation (along with Kepler) later in the course.

For Galileo:

http://galileo.rice.edu/

http://galileo.rice.edu/bio/index.html

I also recommend "Galileo's Daughter" by Dava Sobel. Actually, anything she writes is pretty great historical reading. See also her "Longitude."

It is also worth reading about Copernicus and the Scientific Revolution.

For those of you interested in ancient science, David Lindberg's "Beginnings of Western Science" is amazing.

In general, John Gribbin's "The Scientists" is a good intro book about the history of science, in general. I recommend this for all bio and chem majors.

As a science major, you owe it to yourself to find out the history of your discipline. I think it will give you new perspective and respect.

Test 1 review book pages

Session 7 - Newton and his laws.

Newton, Philosophiae naturalis principia mathematica (1687) Translated by Andrew Motte (1729)

Lex. I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.


Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.


Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.


Lex. II. Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectam qua vis illa imprimitur.


The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.


If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.


Lex. III. Actioni contrariam semper & aequalem esse reactionent: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi.


To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.


Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other.