Foundations of Motion: Newton's Laws, Translational and Rotational Motion, and Applications

Motion and Newton's Laws

Introduction to Motion

Understanding motion is fundamental to physics. The study of motion begins with Newton's three laws, which form the basis of classical mechanics. These laws are derived from careful observation of the natural world.

• Newton's Laws of Motion describe how objects move and interact with forces.

• They apply to both translational (straight-line) and rotational (circular) motion.

Newton's First Law (Law of Inertia)

Definition: A body remains at rest or in a state of uniform motion in a straight line unless acted upon by an external force.

• This law introduces the concept of inertia, the tendency of objects to resist changes in their state of motion.

• Example: A hockey puck slides on ice until friction (an external force) slows it down.

Newton's Second Law (Law of Acceleration)

Definition: The rate of change of linear momentum of a body is equal to the net force applied to it.

• Mathematically, this is expressed as:

$F = ma$

• Where F is the net force, m is mass, and a is acceleration.

• Allows calculation of acceleration if force and mass are known.

• Gravitational force near Earth's surface: $F_{gravity} = mg$, where $g \approx 9.8\ \mathrm{m/s^2}$.

• Example: A 2 kg object under a 10 N force accelerates at $a = F/m = 5\ \mathrm{m/s^2}$.

Newton's Third Law (Action-Reaction)

Definition: For every action, there is an equal and opposite reaction.

• When object A exerts a force on object B, B exerts an equal and opposite force on A.

• Example: When you jump, your feet push down on the ground, and the ground pushes you upward with equal force.

Types of Motion

Translational Motion

In translational motion, all parts of a body move with the same velocity and acceleration in a straight line.

• Example: A car driving straight down a road.

Rotational Motion

In rotational motion, such as a bar rotating around a pivot, all parts of the body rotate through the same angle in the same time, but their linear velocities and accelerations depend on their distance from the center of rotation.

• Example: A spinning wheel or a rotating door.

Combined Motion

Many real-world motions are combinations of translation and rotation, such as a gymnast flipping through the air.

Kinematics: Equations of Motion

Equations for Constant Acceleration

For objects moving with constant acceleration, the following equations apply:

• Final velocity: $v = v_0 + at$

• Acceleration: $a = (v - v_0)/t$

• Average velocity: $v_{av} = (v + v_0)/2$

• Displacement: $s = v_0 t + \frac{1}{2} a t^2$

• Velocity-squared relation: $v^2 = v_0^2 + 2a s$

Where $v_0$ is initial velocity, $v$ is final velocity, $a$ is acceleration, $t$ is time, and $s$ is displacement.

Applications: The Physics of the Vertical Jump

Forces in a Vertical Jump

When a person jumps vertically, two main forces act:

• Weight (W): Downward force due to gravity.

• Reaction Force (F): Upward force exerted by the ground (action-reaction pair).

The net upward force is $F - W$.

Calculating Jump Height

• Let $c$ be the distance the center of gravity is lowered during crouch.

• Acceleration during push-off: $a = (F - W)/m = (F - W)/(W/g)$

• Take-off velocity: $v^2 = 2a c$

• At maximum height, velocity is zero. Using energy or kinematics, the jump height is:

$H = \frac{(F - W)c}{W}$

• If $F = 2W$ (force is twice the weight), then $H = c$.

• For $c = 0.6$ m (average person), $H \approx 0.6$ m.

Energy Considerations

• Work done by muscles: $W_{ext} = F \cdot c$

• At peak, all kinetic energy is converted to potential energy: $F c = W (c + H)$

• Solving for $H$ gives the same result: $H = \frac{(F - W)c}{W}$

Effect of Gravity

• Weight depends on the gravitational acceleration of the planet.

• On the Moon ($g_{moon} \approx g_{earth}/6$), a person can jump much higher for the same force.

Energy Consumption Through Exercise

• Muscles convert chemical energy from food into mechanical work, but efficiency is low (typically <20%).

• Example: A 70 kg person jumping 0.6 m high, once per second for 15 minutes:

• Work per jump: $70 \times 9.8 \times 0.6 = 411$ J

• Total work in 900 jumps: $411 \times 900 = 369,900$ J (370 kJ)

• Only 20% of consumed energy is converted to work; the rest is lost as heat.

Angular Motion and Forces on Curved Paths

Centrifugal and Centripetal Forces

When an object moves in a circle, it experiences a centrifugal force (apparent, outward) and requires a centripetal force (real, inward) to stay on the path.

• Centrifugal force for a car of weight $W$ on a curve of radius $R$:

$F_c = \frac{mv^2}{R} = \frac{W v^2}{g R}$

• Friction provides the centripetal force. Skidding occurs when $F_c$ exceeds frictional force $\mu W$.

• Maximum safe speed: $v_{max} = \sqrt{\mu g R}$

• Banking the road increases safe speed by providing additional centripetal force.

Forces on a Runner on a Curved Track

• Runner leans inward to balance forces.

• Forces on the foot: upward (supports weight) and centripetal (counteracts centrifugal force).

• Resultant force acts at an angle to the vertical.

Pendulum Motion and Simple Harmonic Motion

The Simple Pendulum

A simple pendulum consists of a mass attached to a string, swinging under gravity. Its motion is an example of simple harmonic motion (SHM).

• Period (T): Time for one complete swing.

• Frequency (f): Number of swings per second. $T = 1/f$

• For small angles:

$T = 2\pi \sqrt{\frac{l}{g}}$

• Where $l$ is the length of the pendulum, $g$ is gravitational acceleration.

Energy in a Pendulum

• At the extremes, energy is all potential; at the center, all kinetic.

• Maximum tangential acceleration: $a_{max} = 4\pi^2 A / T^2$ (where $A$ is amplitude).

Walking and the Pendulum Model

• Walking can be analyzed as a series of pendulum-like swings of the leg.

• Example: 120 steps/min (2 steps/sec), step length 90 cm, speed $v = 1.8$ m/s.

• Maximum foot velocity is about three times the body's speed.

The Physical Pendulum

• Unlike the simple pendulum, the physical pendulum accounts for mass distribution.

• Period: $T = 2\pi \sqrt{\frac{I}{mgd}}$

• Where $I$ is the moment of inertia about the pivot, $m$ is mass, $g$ is gravity, $d$ is distance from pivot to center of mass.

• For a uniform rod pivoted at one end , $I=\frac13mL^2,where$L is the rod's length.

• For a rectangular plate pivoted at a corner, I = $\frac{1}{12}m\left(l^2+w^2\right),where$l and w are the length and width of the plate.

Example : A uniform rod pivoted at one end

A rod of mass 0.5kg and length 1m is pivoted at one end. The distance to the centre of mass is 0.5m.

STEP 1 : Calculate the moment of inertia:

I = $\frac13mL^2=\frac13\left(0.5\operatorname{kg}\right)\left(1m\right)^2=0.167\operatorname{kg}\cdot m^2$

$T=2\pi\sqrt{\frac{0.167\operatorname{kg\cdot m^2}}{\left(0.5\operatorname{kg}\right)\left(9.8ms^{-2}\right)\left(0.5m\right)}}=1.64$seconds.