Comprehensive Guide to Physics: Motion, Forces, and Energy

Posted on Aug 26, 2024 in Physics

Newton’s Laws of Motion

Newton’s First Law

When an unbalanced force with an acceleration of zero acts on an object, it will either remain stationary or move in a straight line at a steady speed.

Newton’s Second Law

When an unbalanced force acts on an object with mass, it causes the object to accelerate.

Momentum

Definition

Momentum is the product of an object’s mass and velocity. An unbalanced force acting on an object causes a change in velocity, thus changing the momentum of the object.

Units: kgms^-1

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it.

Energy

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed but can be transformed from one form to another.

• Elastic Collision: Total energy remains unchanged.
• Inelastic Collision: There is a loss of energy.

Impulse

Definition

Impulse is the change in momentum of an object.

Units: kgms^-2 (or Ns)

Banking

Banking a road or track increases the centripetal force, allowing vehicles to navigate curves at higher speeds. The centripetal force is provided by the horizontal component of the reaction force from the banking and the friction force between the road and the vehicle.

• Increasing the reaction force increases the horizontal friction force, resulting in a greater centripetal force and a higher maximum speed for navigating a bend.

• In an isolated system, momentum remains constant.
• If the masses do not change, the velocity of the center of mass remains the same.

Circular Motion

As an object moves around a circle, the direction of its motion changes, and therefore its velocity changes, meaning it is accelerating.

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path. This change in velocity is caused only by a change in direction.

Centripetal Force

Centripetal force is the unbalanced force that causes centripetal acceleration.

Types of Circular Motion

Horizontal Circular Motion

Motion in a horizontal circle, such as a car turning on a flat road.

Vertical Circular Motion

Motion in a vertical circle, such as a roller coaster looping the loop.

Newton’s Law of Universal Gravitation

Newton’s Law of Universal Gravitation states that any two objects are attracted by a gravitational force (F_g). The magnitude of this force depends on:

• Masses of the objects: The gravitational force is directly proportional to the mass of each object.
• Distance between the objects: The force is inversely proportional to the square of the distance (r) between the objects.

Orbital Motion

Motion of an object around another object due to gravitational attraction.

Types of Motion

Pure Translation

Pure translation is the movement of the center of mass of an object without any rotation about the center of mass.

Pure Rotation

Pure rotation is a circular movement around the center of mass, while the center of mass itself remains stationary.

Angular Displacement

• Symbol: Θ
• Units: Radians
• Definition: The angle something turns through in a given time.

Angular Velocity

The rate of change of angular displacement.

Angular Acceleration

• Definition: The rate of change of angular velocity.
• Units: rads^-2

Relationship between linear acceleration (a) and angular acceleration (α):

Torque

The turning effect of a force is called torque (measured in Nm).

• The magnitude of the torque depends on the size of the force and the perpendicular distance from the force to the center of rotation.

Rotational Inertia

Rotational inertia (or moment of inertia) is a measure of an object’s resistance to changes in its rotational motion.

Rotational inertia depends on:

• Mass of the object: Increasing the mass increases the rotational inertia.
• Distribution of mass: Mass located farther from the center of rotation increases the rotational inertia.

Angular Momentum

Angular momentum (L) depends on the angular velocity and the rotational inertia (units: kgm²s^-1).

The law of conservation of angular momentum states that the total angular momentum of a system remains constant if no external torque acts on it.

Linear and Rotational Motion Combined

When an unbalanced force acts on an object, unless it acts through the center of mass, the resulting motion will be a combination of linear and rotational motion.

Total kinetic energy (E_k(total)) is the sum of rotational kinetic energy (E_{k(rotational)}) and linear kinetic energy (E_k(linear)):

E_k(total) = E_{k(rotational)} + E_k(linear)

Cyclic and Oscillatory Motion

Cyclic Motion

Cyclic motion is any motion of a system that repeats itself at regular, equal intervals of time.

Oscillatory Motion

Oscillatory motion is cyclic motion in a straight line, such as the motion of a pendulum.

Simple Harmonic Motion (SHM)

In simple harmonic motion (SHM), the acceleration is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.

• At the ends of the oscillation, acceleration is maximum, and velocity is zero.
• At the equilibrium position, displacement, force, and acceleration are zero, and velocity is maximum.

Restoring Force

In SHM, there is always a restoring force acting towards the equilibrium position.

Reference Circle and Phasor

A reference circle is used to visualize SHM. The rotating radius of the reference circle is called a phasor. The length of the phasor represents the amplitude of the SHM.

Displacement, Velocity, and Acceleration in SHM

• Displacement (y): y = A sin ωt
• Velocity (v): v = Aω cos ωt
• Acceleration (a): a = -Aω² sin ωt

(where A is amplitude, ω is angular frequency, and t is time)

Energy in SHM

The amplitude of SHM indicates the energy of the oscillating object. During the cycle of SHM, the energy changes between kinetic energy and potential energy (gravitational or elastic).

Damping

Energy is continually lost from the system as heat due to friction, causing the amplitude to decrease and the SHM to gradually die away.

Forced SHM

Forced SHM occurs when an external force drives a system to oscillate at a specific frequency.