7.2 Newton's Laws, Buoyancy, and Gravity

7.2 Newton's Laws, Buoyancy, and Gravity

Quick Answer: Newton 1: (F_{net}=0) means constant velocity (including rest); more mass means more inertia. Newton 2: (a = F_{net}/m) in the net-force direction. Newton 3: equal-and-opposite forces act on two different objects. An object floats when buoyant force equals weight (relative density < 1). Mass is constant; weight (W=mg) changes with (g). In free fall without air resistance, all masses accelerate at the same (g).

Kinematics describes motion; forces explain changes in motion. For Praxis Middle School Science (5442) II.B.1, you need Newton's three laws, buoyancy and density reasoning, gravity (including weight vs. mass), the vector nature of force, and projectile/free-fall ideas with air resistance neglected. Expect both calculation-light quantitative items and teaching scenarios about student misconceptions.

Force as a Vector

A force is a push or pull. Forces are vectors: they have magnitude (newtons, N) and direction. The net force (F_{net}) is the vector sum of all forces on an object.

Worked example: A box is pulled right with 25 N and left with 10 N on a frictionless surface. (F_{net} = 15,\text{N right}). Direction of acceleration matches the net force.

Newton's First Law — Inertia

An object remains at rest, or continues moving at constant velocity, unless acted on by a nonzero net external force.

Inertia is the tendency to resist changes in motion; more mass → more inertia.

Applications Praxis likes

• A puck on nearly frictionless ice keeps sliding at nearly constant velocity (net force ≈ 0).
• Passengers lurch forward when a bus brakes: their bodies continue moving at the earlier velocity until a seatbelt exerts a net force.
• A book on a table is not "force-free" — gravity and the normal force cancel, so (F_{net} = 0).

Misconception to correct: "Forces keep objects moving." Actually, net force changes velocity; no net force means constant velocity, not necessarily "stopped."

Newton's Second Law — (F_{net} = ma)

The acceleration of an object is proportional to the net force and inversely proportional to mass, in the direction of the net force.

[ a = \frac{F_{net}}{m} \quad \text{or} \quad F_{net} = ma ]

Worked example

What net force accelerates a 4 kg cart at 3 m/s²?

(F_{net} = ma = 4 \times 3 = \mathbf{12,N}) in the acceleration direction.

Worked example — unbalanced forces

Two students push a 20 kg crate. One pushes with 50 N east; friction and the other forces leave a net force of 30 N east. Acceleration:

(a = 30/20 = \mathbf{1.5,m/s^2\ east}).

Newton's Third Law — Action-Reaction

When object A exerts a force on object B, object B exerts an equal-magnitude, opposite-direction force on A.

Critical details for exam items:

1. The two forces act on different objects.
2. They are equal even if the objects have different masses (the less massive object gets the larger acceleration).
3. They occur simultaneously — not "action then reaction."

Applications

• Rocket: expels gas downward; gas pushes rocket upward (works in vacuum — exhaust does not need air to "push against").
• Walking: foot pushes ground backward; ground pushes foot forward.
• Swimmer: hands push water backward; water pushes swimmer forward.

Misconception: "If forces are equal and opposite, nothing can accelerate." Equal-and-opposite third-law forces do not cancel because they are not both on the same free-body diagram.

Buoyancy and Relative Density

Buoyant force is the upward force a fluid exerts on an immersed object (Archimedes: equal to the weight of the displaced fluid).

Relative density (specific gravity) ≈ density of object ÷ density of fluid.

Worked example — sink or float

Wood density ≈ 0.7 g/cm³; water ≈ 1.0 g/cm³ → relative density 0.7 < 1 → floats. Iron ≈ 7.9 g/cm³ → relative density 7.9 > 1 → sinks in water (but a steel ship floats because its average density, including air-filled hull, is less than water).

Teaching cue: Students may say "heavy things sink." Counterexample: a massive log can float; a tiny pebble sinks. Compare densities (or average density), not weight alone.

Gravity, Mass, Weight, and Distance

Newton's law of universal gravitation (conceptual level for middle school / Praxis):

[ F_g \propto \frac{m_1 m_2}{r^2} ]

Gravitational force is stronger when masses are larger and weaker when center-to-center distance (r) increases (inverse-square).

On Earth, (g \approx 9.8,\text{m/s}^2) (often ~10 m/s² for estimation). On the Moon, (g_{Moon} \approx 1.6,\text{m/s}^2) — roughly 1/6 of Earth's.

Worked example — Earth vs. Moon weight

A student has mass 60 kg.

• Weight on Earth ≈ (60 \times 9.8 = \mathbf{588,N})
• Weight on Moon ≈ (60 \times 1.6 = \mathbf{96,N})
• Mass on Moon remains 60 kg

Exam trap: "Would you weigh less on the Moon?" Yes. "Would your mass change?" No.

If Earth–object distance increases (for example, moving far above Earth's surface), gravitational force decreases approximately with (1/r^2). Doubling distance from Earth's center (not from the surface) reduces gravitational force to about one-fourth, all else equal.

Free Fall and Mass Independence

When air resistance is neglected, the only force on a falling object near Earth is gravity (weight). Then:

[ a = \frac{W}{m} = \frac{mg}{m} = g ]

All masses accelerate at the same rate (g) in free fall (Galileo's insight; Apollo hammer-and-feather demo on the Moon). A 1 kg ball and a 5 kg ball dropped from the same height hit the ground together in vacuum (or when air drag is negligible).

With air resistance, shape and speed matter: a flat paper falls slower than a crumpled paper of equal mass because drag is larger relative to weight.

Projectiles with Only Gravity (No Air Resistance)

A projectile is launched and then moves under gravity alone (idealized). Near Earth:

Worked example — conceptual

A ball is thrown horizontally off a table at the same instant an identical ball is dropped from the table height. Neglecting air resistance, both hit the floor at the same time because vertical motion depends only on (g) and initial vertical velocity (both start with (v_{vertical} = 0)). The thrown ball travels farther horizontally because it keeps its horizontal speed.

Worked example — vector force reminder

At every point on the trajectory (ideal case), the force of gravity is straight down. There is no "force of motion" pushing along the path. Velocity can have horizontal and vertical components; the force (and acceleration) stays vertical.

Putting the Laws Together — Classroom Force Catalog

Equilibrium vs. accelerating systems

• Book at rest on table: weight down, normal up, (F_{net} = 0) (first law).
• Book sliding at constant speed while pushed: applied force balances friction; (F_{net} = 0) even though forces are present (first law again).
• Book speeding up: applied force exceeds friction; (F_{net} \neq 0); (a = F_{net}/m) (second law).

Teaching-Scenario Highlights for 5442

About 30% of items are instructional. Common stems:

1. Student says a rocket moves because exhaust "pushes on air" — correct focus is third-law interaction with expelled gas (works in vacuum).
2. Student says heavier objects fall faster — counter with free-fall independence of mass (neglect air) or equal-acceleration demos.
3. Student says floating depends on being "light" — redirect to density / buoyant force vs. weight.
4. Student says no forces act on a coasting hovercraft — redirect to net force of zero, not "no forces."

Connect back to 7.1: forces cause acceleration (change in velocity); constant velocity means net force zero, which is why v–t graphs go horizontal when balanced forces appear.