Physics 8 — Wednesday, October 5, 2011

I handed out HW #4 and HW #5 on Monday. Pick them uptoday if you were not here on Monday. (You can also find them onthe course web page.)

Chapter 8 (Force) — part two

“The interaction between two objects involves equal and oppositeforces. Every object that exerts a force on another object will havean equal and opposite force exerted on it by that same object.This is described in Newton’s third law of motion. For example, if Islap a table, not only am I exerting a force on the table with myhand, but the table would exert an equal and opposite force on myhand (thus causing some degree of pain in my hand).”

“Where there is an action there is always a reaction. When I’mpushing a table, the table is also pushing on me. When I’m liftinggrocery bags off the floor, they are exerting a pull down on myhands. A locomotive is pushing the wagons but the wagons areexerting an equal force on the locomotive.”

“This third law of motion is an expression of the conservation ofmomentum in terms of forces.”

“When two objects come into contact, the first object exerts aforce on the second object that is equal in magnitude and oppositein direction to the force exerted on the first object by the secondobject. An everyday example of this would be a book resting on atable. The force of the book on the table is equal and opposite tothe force of the table on the book.”

“An everyday example, is when I lean up against a door to lookcool — because the door is holding me up with the samemagnitude and in the opposite direction, otherwise I would falldown and then I wouldn’t be so cool.”

“Relating force to conservation of momentum is so helpful; I don’tthink I thought about this in high school physics.”

“I just really enjoy saying ‘equal and opposite direction.’ ”

“If a system has no external force, the center of mass will remainat rest or moving at constant velocity if it is already moving. Ifthere is an external force, the center of mass accelerates accordingto ~F = m~a. The center of mass can be treated as a point mass,following Newton’s Laws.”

“If external forces are exerted on the objects in a many-objectsystem the system’s center of mass moves as though all the objectsin the system were concentrated at the center of mass and allexternal forces exerted at that point.”

“Yes, the concept of center of mass seems more useful. It seems tobe related to forces in the idea of external vs internal forces. Whenthere are no external forces in a system, the system’s center ofmass moves at a constant velocity. When there are no externalforces on a system, just internal forces, it seems that a system’scenter of mass would remain constant. Perhaps this is because ofthe cancellation of internal forces — and the idea that internalforces always add to zero.”

“External forces affect a system’s center of mass velocity. Theinternal contact forces in a system do not affect this velocity sincethe forces cancel each other out, since they are equal in magnitudeand opposite in direction.”

“center of mass is very useful concept because this concept isinterlinked with momentum and the force. in case of force, ifexternal forces are exerted on the objects in a many-object systemthe system’s center of mass moves as though all the objects in thesystem were concentrated at the center of mass and all externalforces exerted at that point.”

“I think the organization of the book is excellent — starting withconservation laws then transitioning to forces helps you betterunderstand the concept of force.”

Chapter 8 (Force) — sources of confusion

I Why is Hooke’s law always linear?

I What is impulse?

I Equation of motion

I Drawing free-body diagrams

I Be sure not to confuse the “equal and opposite” of interactionpairs with the balancing of forces required to keep an objectmoving at a constant (possibly zero) velocity!

I Remember that external forces act on a system’s center ofmass.

Equal and opposite forces?A locomotive pulls a series of wagons. Which is the correctanalysis of the situation?

(A) The train moves forward because the locomotive pulls forwardslightly harder on the wagons than the wagons pull backwardon the locomotive.

(B) Because action always equals reaction, the locomotive cannotpull the wagons-the wagons pull backward just as hard as thelocomotive pulls forward, so there is no motion.

(C) The locomotive gets the wagons to move by giving them a tugduring which the force on the wagons is momentarily greaterthan the force exerted by the wagons on the locomotive.

(D) The locomotive’s force on the wagons is as strong as the forceof the wagons on the locomotive, but the frictional force onthe locomotive is forward and large while the backwardfrictional force on the wagons is small.

(E) The locomotive can pull the wagons forward only if it weighsmore than the wagons.

Tropicana juice train really exists!

Let’s see the effect of including or not including thefrictional force of the tracks pushing forward on thewheels of the engine.

I’ll pretend to be the engine!

External forces act on system’s CoM

Let’s define “system” to be engine+wagon.Remember that forces internal to system cannotaccelerate system’s CoM.

To change the velocity of the CoM, we need a forcethat is external to the system.

Equal and opposite forces?

Consider a car at rest. We can conclude that the downwardgravitational pull of Earth on the car and the upward contact forceof Earth on it are equal and opposite because

(A) the two forces form an interaction pair.

(B) the net force on the car is zero.

(C) neither of the above

Let’s draw the free-body diagram for a book sitting on a table —looking at the book and then looking at the table.

Be careful not to confuse the “equal and opposite” of interactionpairs with the balancing of forces required to keep an objectmoving at a constant (possibly zero) velocity!

External force (gravity) acting on CoM

Gravity / CoM demo (quickly)

Hooke’s law

I When you pull on a spring, it stretches

I When you stretch a spring, it pulls back on you

I When you compress a spring, it pushes back on you

I For an ideal spring, the pull is proportional to the stretch

I Force by spring, on load is

Fx = −k (x − x0)

Let’s look at some examples of springs.

Why is Hooke’s law linear? Is it always?

This is easier to answer nextweek. But if you took highschool physics, you learnedUspring = 1

2k(x − x0)2

You can Taylor expandarbitrary U(x) as

U(x0)+(x−x0)U ′(x0)+1

2(x − x0)2U ′′(x0)+

1

6(x−x0)3U ′′′(xo)+· · ·

U(x0) is arbitrary. U ′(x0) = 0 at equilibrium. Cubic term may beignored if we’re not too far from x0. Quadratic term remains.

Tension vs. compression

I When a force tries to squish a spring, that is calledcompression, or a compressive force

I When a force tries to elongate a spring, that is called tension,or a tensile force

I We’ll spend a lot of time next month talking aboutcompression and tension in columns, beams, etc. — very bigdeal

I For now, remember that tension is the force trying to pullapart a spring, rope, etc., and compression is the force tryingto squeeze a post, a basketball, a mechanical linkage, etc.

Tension in cables

I A large category of physics problems involves two objectsconnected by a rope, a cable, a chain, etc.

I These things can pull but can’t push.

I Usually they are considered light enough that you don’t worryabout their inertia (we pretend m = 0), and stiff enough thatyou don’t worry about their stretching when you pull on them(we pretend k =∞).

I The cable’s job is just to transmit a force from one end to theother. We call that force the cable’s tension, T .

I Cable always pulls on both ends with same magnitude (T ),though in opposite directions

I E.g. hang basketball from ceiling. Cable transmits mg toceiling. Gravity pulls ball down. Tension pulls ball up. Forceson ball add to zero.

Tension in cables . . .

In the 17th century, Otto von Guricke, a physicist in Magdeburg,fitted two hollow bronze hemispheres together and removed the airfrom the resulting sphere with a pump. Two eight-horse teamscould not pull the halves apart even though the hemispheres fellapart when air was readmitted. Suppose von Guricke had tied bothteams of horses to one side and bolted the other side to a heavytree trunk. In this case, the tension on the hemispheres would be

(A) twice

(B) exactly the same as

(C) half

what it was before.