Physics Chapter on Significant Figures and Motion

Significant Figures

• Significant figures are digits that contribute to the precision of a number.
• Leading zeros in a number with a decimal point are not significant.
• Trailing zeros in a number with a decimal point are significant.

Dimensional Formulae

• The dimensional formulae for physical quantities are as follows:
  • Force: MLT⁻²
  • Work: ML²T⁻²
  • Power: ML²T⁻³

Kinematic Equations of Motion

• The kinematic equations of motion for an object under free fall are:
  • v = u + gt
  • s = ut + ½gt²
  • v² = u² + 2gs
  • where: v = final velocity, u = initial velocity, g = acceleration due to gravity, t = time, and s = displacement.

Horizontal Range of a Projectile

• The horizontal range of a projectile is the horizontal distance traveled by the projectile before it hits the ground.
• The horizontal range is given by:
  • R = (u² sin 2θ) / g
  • where: u = initial velocity, θ = angle of projection, and g = acceleration due to gravity.

Newton’s Third Law of Motion

• Newton’s third law of motion states that for every action, there is an equal and opposite reaction.
• This means that when two objects interact, they exert forces on each other that are equal in magnitude and opposite in direction.
• Example: When you push a wall, the wall pushes back on you with an equal and opposite force.

Work-Energy Theorem

• The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
• This means that if you do work on an object, you are giving it energy.
• The work-energy theorem can be expressed mathematically as:
  • W = ∆KE
  • where W is work done and ∆KE is the change in kinetic energy.

Linear Motion vs. Rotational Motion

• Linear motion is the motion of an object along a straight line.
• Rotational motion is the motion of an object around a fixed axis.
• Some key differences include:
  • Linear motion is characterized by displacement, velocity, and acceleration, while rotational motion is characterized by angular displacement, angular velocity, and angular acceleration.
  • Linear momentum is conserved in linear motion, while angular momentum is conserved in rotational motion.

Total Energy of an Orbiting Satellite

• The total energy of an orbiting satellite around the Earth is the sum of its kinetic energy and potential energy.
• This total energy is constant and is given by:
  • E = -GMm / (2r)
  • where: G = gravitational constant, M = mass of the Earth, m = mass of the satellite, and r = orbital radius.

Period of Revolution of a Satellite

• The period of revolution of a satellite very close to the surface of the Earth is given by:
  • T = 2π√(RE/g)
  • where: RE = radius of the Earth and g = acceleration due to gravity.

Velocity-Time Graph

• A velocity-time graph is a graph that plots the velocity of an object as a function of time.
• The slope of a velocity-time graph represents the acceleration of the object.
• The area under a velocity-time graph represents the displacement of the object.

Centripetal Acceleration

• Centripetal acceleration is the acceleration that is required to keep an object moving in a circular path.
• It is always directed towards the center of the circle.
• The magnitude of centripetal acceleration is given by:
  • ac = v²/r
  • where: v = velocity, and r = radius of the circular path.

Laws of Kinetic Friction

• Friction is a force that opposes motion between two surfaces in contact.
• Kinetic friction is the friction that occurs when two surfaces are sliding past each other.
• Laws of kinetic friction state:
  • The kinetic friction force is proportional to the normal force acting between the two surfaces.
  • The kinetic friction force is independent of the area of contact between the surfaces.
  • The kinetic friction force is independent of the relative speed of the surfaces.

Methods of Reducing Friction

• Friction can be reduced by:
  • Using lubricants: a substance applied between two surfaces that reduces friction by creating a thin layer of separation.
  • Making surfaces smoother: smoother surfaces have less friction.
  • Using ball bearings or roller bearings: reduces friction by rolling instead of sliding.
  • Using air or fluid bearings: creates a thin layer of air or fluid between the surfaces that reduces friction.

Conservation of Mechanical Energy

• The law of conservation of mechanical energy states that the total mechanical energy of a system remains constant if no external forces act on the system.
• This means that the sum of the kinetic energy and potential energy of an object remains constant as long as no work is done by external forces.
• This can be applied to a freely falling body by examining the potential and kinetic energy at different points during the fall.

Angular Momentum

• Angular momentum is a measure of the amount of rotational inertia of a body.
• It is a vector quantity and its direction is perpendicular to the plane of rotation.
• The magnitude of angular momentum is given by:
  • L = Iω
  • where: I = moment of inertia, and ω = angular velocity.

Time Rate of Change of Angular Momentum

• The time rate of change of angular momentum is equal to the net torque acting on the body.
• This can be expressed mathematically as:
  • dL/dt = τ

Escape Velocity

• Escape velocity is the minimum velocity that an object needs to escape the gravitational pull of a planet or star.
• This means that if an object is launched with escape velocity from the Earth’s surface, it will never fall back to Earth.
• The escape velocity of an object from the Earth is derived from the equation:
  • ve = √(2GM/R)
  • where: G = gravitational constant, M = mass of the Earth, and R = radius of the Earth.

Magnitude and Direction of Resultant Vector

• The magnitude of the resultant of two vectors A and B is given by:
  • |R| = √(A² + B² + 2ABcosθ)
  • Where: θ is the angle between A and B.
• The direction of the resultant vector can be determined using the law of cosines.

Newton's Second Law of Motion

• Newton's second law of motion states that the rate of change of momentum of a body is directly proportional to the force applied to it, and the change occurs in the direction of the applied force.
• This means that the larger the force that is applied to an object, the larger its acceleration will be.
• If we derive this into an equation we find that the force (F) is equal to the mass (m) of the object times its acceleration (a)
  • F = ma

Instantaneous Power

• Instantaneous power is the rate at which work is done at a particular instant in time.
• The instantaneous power is given by:
  • P = dW/dt
  • where: P = power, W = work done, and t = time.

Commercial Unit of Electrical Power

• The commercial unit of electrical power is the kilowatt-hour (kWh).
• One kilowatt-hour is equal to the amount of energy used by a 1000 watt appliance for one hour.

Force, Work, and Power

• The SI units of force, work, and power are:
  • Force: Newton (N)
  • Work: Joule (J)
  • Power: Watt (W)
• The relationships between them in terms of dimensions are:
  • Force = MLT⁻²
  • Work = ML²T⁻²
  • Power = ML²T⁻³
• The relationships between them in terms of units are:
  • 1 J = 1 Nm
  • 1 W = 1 J/s = 1 Nm/s