Force, Newton’s laws of Motion, gravitation

Force and Motion

Introduction to Force

• In the previous chapter, motion was studied.
• This chapter focuses on the cause of motion: force.
• Philosophers believed continuous force was needed for continuous motion.
• It was known that force could initiate or halt motion.
• The precise meaning of force was unclear to scholars of the past.
• They formulated theories but didn't verify them through experimentation.

Galileo's Experimental Approach

• Galileo Galilei pioneered the study of motion using experiments.
• He used simple tools like clocks to study motion with precision.
• Mathematics was not well-developed, and equations were not commonly used at the time.
• Galileo formulated rules using ratios and proportions.
• For uniform motion, he stated, 'distance covered and time taken are directly proportional.'
• Galileo's work significantly improved the understanding of motion concepts.
• He studied linear, oscillatory, and projectile motion.

Sir Isaac Newton's Contributions

• Sir Isaac Newton (1642-1727) was an influential physicist, mathematician, astronomer, philosopher, alchemist, and theologian.
• He is considered one of history's greatest and most influential scientists.
• His book ‘Philosophiæ Naturalis Principia Mathematica’ (1687) laid the groundwork for classical mechanics.
• Newton described universal gravitation and the three laws of motion.

Newton’s Laws of Motion

• Newton's laws relate forces acting on a body to its motion.
• These laws can be summarized into three statements.

First Law: The Law of Inertia

• A body's velocity remains constant unless acted upon by an external force.
• Inertia is the property of bodies to resist changes in their state of motion.
• A body at rest stays at rest, and a body in uniform motion stays in uniform motion unless a force acts upon it.
• Example: A marble rolling on a frictionless surface would continue to roll forever.
• In reality, friction between the marble and the surface causes it to stop.
• Inertia is directly proportional to mass: more mass implies more inertia.
• Examples of inertia in daily life:
  • A book on a table stays there unless an external force moves it.
  • Passengers in a starting bus feel thrown backward due to inertia of rest.
  • Passengers in a stopping bus are thrown forward due to inertia of motion.
  • Dust is removed from a carpet beaten with a stick because dust particles resist the change in motion.
  • A cyclist continues moving for a while after pedaling stops.
  • A person jumping from a moving vehicle must run to avoid falling.

Second Law: Rate of Change of Momentum

• The rate of change of momentum is proportional to the external unbalanced force, and the change occurs in the force's direction.
• The mass is assumed constant during the force application.
• Formula: Force = \frac{change \ in \ momentum}{time}
• F = \frac{mv - mu}{t}, where:
  • m = mass,
  • v = final velocity,
  • u = initial velocity,
  • t = time.
• F = m \frac{(v-u)}{t}
• F = ma, since \frac{(v-u)}{t} = a (acceleration).
• The unit of force is the Newton (N).
• 1 Newton is the force that produces an acceleration of 1 m/s^2 when applied to a mass of 1 kg.
• The unit of force in the CGS system is the dyne.
• 1 dyne is the force that produces an acceleration of 1 cm/s^2 when applied to a mass of 1 g.

Third Law: Action and Reaction

• For every action (force), there is an equal and opposite reaction (force).
• Action and reaction forces act on different bodies.
• Examples:
  • A rocket expels hot gases backward (action), and the gases push the rocket forward (reaction).
  • A swimmer pushes water back (action), and water pushes the swimmer forward.
  • While walking, we push the ground back (action), and the ground pushes us forward.
  • When a bullet is fired from a gun, the gun recoils backward.

Newton’s Law of Gravitation

• Every point mass attracts every other point mass in the universe.
• The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Formula: F \propto \frac{m1 m2}{r^2}
• F = G \frac{m1 m2}{r^2}, where:
  • m1 and m2 are the masses,
  • r is the distance between the masses,
  • G is the universal gravitational constant.
• The value of G was measured by Henry Cavendish in 1798: G = 6.67 \times 10^{-11} Nm^2/kg^2.

Gravitational Acceleration

• When an object is dropped, Earth applies gravitational force, accelerating it toward the ground.
• This acceleration is called gravitational acceleration ('g').
• The value of 'g' depends on the distance from the Earth's center.
• By Newton's second law: F = ma
• Gravitational force = mass x gravitational acceleration.
• G \frac{m1 mE}{r^2} = m \times g, where:
  • m_E is the mass of the Earth,
  • r is the distance between the object and the Earth's center.
• g = G \frac{m_E}{r^2}
• Substituting m_E = 5.98 \times 10^{24} kg and r = 6.37 \times 10^6 m, we get g = 9.8 m/s^2 on Earth.
• 'g' depends on the mass and radius of the Earth but not on the object's mass.
• All objects experience the same gravitational acceleration.
• A larger and smaller stone dropped from the same height will reach the ground at the same time.
• The Earth's radius is greater at the equator than at the poles, so 'g' varies:
  • g(equator) = 9.780 m/s^2
  • g(pole) = 9.832 m/s^2

Mass vs. Weight

• Mass is the quantity of matter in an object.
• All objects have mass, measurable in kg.
• Weight is the gravitational force exerted on an object by Earth.
• Weight is a force and is measured in Newtons.
• Weight depends on the object's distance from the Earth's center and can vary.
• Mass remains constant regardless of location.
• Weight = gravitational force between object and Earth.
• W = G \frac{m1 mE}{r^2}, But we know that g = G \frac{m_E}{r^2}
• Hence w = mg, where g = 9.8 m/s^2.
• Weight (in Newtons) = mass (in kg) x 9.8.
• The weight of a 1 kg mass is 9.8 N.
• Saying