Physci text 05 Forces

Newton's Laws of Motion

Newton's first law: When the momentum is a constant including zero

p = constant

Newton's first law is a logical extension of the concepts of the conservation of momentum. Newton's first law simply says that the momentum remains the same. The variable used for momentum is p. Newton's first law is usually applied to single objects where the mass does not change. In this case the velocity of the object stays the same. If the object is at rest, then the object remains at rest. If the object is moving, the the object continues to move. The extension is that the law notes that the motion remains constant provided no external forces act on the object. As Newton said:

Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by unbalanced forces impressed on it. - Isaac Newton, Philosophaei Naturalis Principia Mathematica

The tendency of an object at rest to remain at rest, or if in motion to stay in motion, is also referred to as inertia. Inertia can be thought of as the natural resistance an object has to changing its state of motion.

Note that a single mathematical expression p = constant includes both rest (constant = 0) and motion (constant ≠ 0). This "symmetry" is more apparent when one is in constant, unaccelerated motion alongside a moving object. When I run next to a car moving at the same speed as I am running, the car appears motionless relative to me. I would say the car has a momentum of zero. A child sitting in a nahs would say both the car and I have a non-zero momentum.

The somewhat amazing aspect of momentum is that in collisions such as the marble collisions, all observers would say momentum is conserved. Both those sitting at the table and those moving with the speed of the rolling marble. The later observer would say the line of marbles moved backward into the one marble, collided, and "left one behind" stationary relative to the moving observer. If your head hurts a little thinking about this, do not worry, this is normal!

Newton's second law: When the momentum is a changing

Δp = constant

Note the ∆, this means "the change in." There was no change in momentum in the first law. Now there is a change, a change at a constant rate.

Newton's second law builds on the first law. When the momentum changes with respect to time at a constant rate, this rate is called a force. What force is at a basic, fundamental level and how a force comes into existence remains difficult to characterize. Quantum field theory invokes particles to explain force, but that does not help us mere mortals who do not understand quantum field theory understand what a force is at a basic level.

In part one of laboratory four when one marble was rolled into the chain of marbles the force of the impact of the rolling marble was transmitted down the line of marbles to the last marble which then carried the original momentum away from the collision. Just as the "one-in, one out," "two-in, two-out," and "three-in, three out" was in some way mysterious, so too are forces.

Force = Δp ÷ Δt = Δmomentum ÷ Δtime

Newton himself thought of force as arising out of a change in momentum. Newton used the term quantity of motion for what we now call the momentum p. In Newton's words:

The net force acting on a body is equal to the rate at which the body's quantity of motion is changing.

As was the case with Newton's first law, the mass is often considered to remain the same. Considering mass to be constant and remembering that the momentum p is equal to the mass times the velocity allows for rewriting the second law in the following way:

Force = Δp ÷ Δt⃗ = Δmv ÷ Δt = m (Δv÷Δt) = ma

Where m is the mass of the object, v the velocity of the object, and a the acceleration of the object.

If a force acts on an object, the object will accelerate. In laboratory three the force of gravity acted on the falling ball. The ball fell faster and faster as it fell. The rate of the increase in speed was the acceleration.

Newton's third law: Force and counter-force

As noted above, force is a complex concept. For an object that is free to move, force can cause the object to move and to accelerate. For an object that is not free to move in the direction of the force, the force can still act on the object even though the object is not moving. The study of the application of force to fixed, non-moving structures is called statics.

Forces act on steel beams, bridge spans, building structures, tires, springs, and many other structures. Those forces may include the force of gravity on a bridge, the force of the wind on a building, or the force of the friction with the road on a tire. In each of these cases a counterforce must counter-act the first force or the object will move. While a moving tire may be desirable, a collapsing bridge or a building falling over are not desirable outcomes.

If I lean on the wall, I exert a force on the wall. If the wall did not push back, then I would fall. Imagine a wall made of water, I would simply fall into the wall of water. Water does not push back.

Force object A on object B = −Force object B on object A

From the above comes the concept of force pairs. For every force that does not produce an acceleration, there is an equal and opposite force acting. This is sometimes said, "For every action there is an equal and opposite reaction." That is not entirely accurate, better to say, "For every force there is an equal and opposite counter-force."

Newton's third law is central to the work done by architects and engineers who use statics to analyze the forces on designed structures to ensure the structures are strong and will not fail.

Instructional notes: Demonstrations have included carrying a ball while coasting on a swizzle board to illustrate frames of reference. When the weather is nice, the class can be wrapped up by a third law demonstration known as a yurt circle outside. The class stands in a large circle and then counts off. The circle must be even, the instructor joins if the class is an odd number. Everyone holds hands or locks wrists. On the count of three those with odd numbers lean in, those with even numbers lean out. Done slowly and carefully, the circle will balance, with everyone off-balance.

Laboratory 05: Force of Friction

Questions being explored

•     What factors contribute to friction: weight, surface area, or surface roughness?

•     Which factor contributes the most?

•     Which factor contributes the least?

•     What is the nature of the relationships between the factors and the force of friction

Equipment

•     sleds of different sizes with strings

•     spring scales

•     mass balance

•     weights

•     sandpaper of various grits across as wide a range as possible.

•     rulers

Instructional note: Broken louvers were originally used to create sleds for this laboratory in Micronesia. These must be handled with care. The upside is that there is no information on the sliding coefficient of glass on sandpaper on line. And the glass does not necessarily behave as might be expected based on information that is on line. The surface area controls for weight by the use of three sleds each consisting of two slabs of glass of different areas glued together with the yoke string in the middle. These three pairs are stacked and flipped in various orders to produce a constant weight sled with different surface areas. In later years wood, PVC board, and tile sleds have been used. Surface area is set aside in some terms.

Data Table

Each group works with one of three tables: weight, grit, or surface area. More recently surface area was dropped with the move to different materials. Tables may have more or less rows depending on the number of variations of a factor available. Tables can be entered directly into Desmos for those students with the ability to do so. This provides real time feedback on any emerging patterns. The tables below are shortened and are shown only as examples. The actual tables may have many more data values

Analysis

Report whether the data is linear, non-linear, or random. If the data is linear, how good is the fit? What is the nature of the relationship: positive or negative? Does the relationship appear to be a strong or weak relationship? If Desmos is being used and the tables are labeled as above, then the regression equations for each would be: f1~m1w1+b1, f2~m2s2+b2, f3~m3g3+b3

The three slopes will be m1, m2, m3. Slopes near zero may indicate no relationship between the variables. The intercepts will be b1, b2, b3. If the slope is NOT close to zero and the intercept IS close to zero, then there may be a positive relationship for the data in this laboratory.

Discussion

Discussion the results of your group and of the other groups in the laboratory report.

Instructional note: At the end of the laboratory the students can be asked to work on a brief presentation for the next day in the class. This could be the next lecture day of the class. This provides an opportunity for the teams of scientists to share their findings and explore which factors had the most effect, which had the least effect. In one class the instructor uses this opportunity to introduce the students to looking at the correlation r value displayed by Desmos.