Mechanics

Static Equilibrium

A condition where the sum of all forces and the sum of all torques acting on a structure are both zero.

Torque

A measure of the rotational force on an object, calculated about a specific axis. Can cause rotation or torsion.

Right Hand Rule

A method used to determine the direction of forces based on the orientation of the fingers of the right hand for the cross product of two vectors.

Vector Cross Product

A mathematical operation used to calculate rotational quantities, defined as $aXb=ab\sin\theta$ .The result is a vector that is perpendicular to the plane formed by the two vectors being multiplied.

Maximum Torque

Occurs when the force is applied perpendicular to the position vector. It quantifies the greatest rotational effect achievable given a specific distance and force applied.

Torque in 3D

Torque is a vector that is perpendicular to the plane defined by the cross product of the position vector and force vector.

Positive Torque

Torque directed along the positive z-axis.

Negative Torque

Torque directed along the negative z-axis.

Parallel Axis Theorem

Allows the calculation of the moment of inertia of an object about any axis parallel to one through its center of mass.

Perpendicular Axis Theorem

States that for a plane object, the moment of inertia about an axis perpendicular to its plane is the sum of the moments of inertia about two axes in its plane.

Angular Velocity (ω)

The rate of change of angular displacement of an object, typically measured in radians per second.

Continuous Mass Distribution

When mass is spread out over a volume rather than concentrated at discrete points.

Compound Object

An object made up of two or more different bodies, each with its own moment of inertia.

Distribution of Mass

The arrangement of mass within an object, affecting its moment of inertia with respect to a chosen axis.

Center of Mass (CoM)

The point in a body or system of bodies where the mass can be considered to be concentrated. Mainly used for calculating quantities for translational motion. It represents the average position of all mass in the system, allowing for simplified analysis of motion.

Conservation of Angular Momentum

If external forces produce no net torque on a system, the angular momentum remains constant.

Angular Momentum

A measure of rotational motion defined as the product of an object's moment of inertia and its angular velocity or the cross product of its position vector and momentum for a rigid body. It quantifies the extent of rotation of an object around an axis and is preserved in a closed system without external torque.

Gyroscope

A device consisting of a spinning disk which is free to assume any orientation, used to demonstrate angular momentum conservation. Gyroscopes maintain their orientation due to angular momentum, making them useful in navigation and stability systems as angular momentum has a constant direction in space.

Precession

The phenomenon where the axis of a spinning object moves in response to an external torque, changing the direction of its angular momentum. This results in a gradual shift of the rotation axis, typically observed in gyroscopes or spinning tops. During precession, the magnitude of angular momentum remains constant while its direction changes.

External Torque

A torque that results from forces applied from outside the system.

Rate of Precession (Ω)

The rate at which the axis of a spinning body precesses around the vertical axis.

Rigid Body

An object with a fixed shape that does not deform under the application of forces, maintaining the distance between any two points.

Newton's Second Law for Rotation

The change in angular momentum of a system is equal to the net external torque applied to it, expressed as $\frac{dL}{dt} = \tau$.

Rotational Kinetic Energy

The kinetic energy of an object due to its rotation, expressed as $K_{r}={}\frac12I\omega^2$ .It is the energy possessed by a body due to its rotational motion about an axis.

Total Kinetic Energy

The sum of rotational and translational kinetic energies, expressed as $K = K_t + K_r$.

Angular Velocity ($\omega$)

The rate of change of the angle with respect to time, typically denoted by \frac{d\theta}{\differentialD t} .

Linear Velocity or Tangential velocity ($v$)

The speed of a point in a rotating object, given by the product of the radius and angular speed: $v_{i}=r_{i}\omega$ .It represents how fast the object is moving along a circular path, where $r$ is the radius and $\omega$ is the angular velocity.

Angular Displacement

The change in the angle (in radians) during rotation, represented as ∆𝜃 = 𝜃(B) − 𝜃(A).

Kinematic Equations for Constant Angular Acceleration

Equations that relate angular displacement, angular acceleration, and angular velocities in rotational motion.

Relationship Between Angular and Translational Quantities

Particles in a rotating rigid body share the same angular displacement, velocity, and acceleration, but translational quantities depend on their radial distance from the axis.

Angular Acceleration

The rate of change of angular velocity of a rigid body due to the sum of external torques.

Work-Energy Theorem

A principle that relates the work done on an object to its change in kinetic energy, used to analyze rotation.

Moment of Inertia (I)

A quantity expressing a body's tendency to resist angular acceleration, dependent on mass distribution.

Power (P) for Rotational Motion

The rate at which work is done in rotational motion, given by P=\tau\frac{d\theta}{\differentialD t}=\tau\omega.

Translational Motion vs. Rotational Motion

Translational motion refers to movement along a path (linear), while rotational motion refers to movement around an axis.

Centre of Mass for multiple particles

Given particles of mass $m_1$ and $m_2$, the CoM can be calculated using the formula $x_C = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$.

Centre of Mass for continuous objects

For an extended body like a wire, the CoM is calculated as $x_c = \frac{\sum x_i \Delta m}{\sum \Delta m}$ where $\Delta m$ is an infinitesimally small mass.

Locating Centre of Mass

The process of finding the CoM involves locating the CoM of each object and treating them as point masses.

Newton’s 2nd Law for a System of Particles

The motion of the centre of mass depends only on the vector sum of all external forces acting on the system.

Conservation of Linear Momentum

A principle stating that the total linear momentum of a closed system remains constant if no external forces act on it.

Elastic Collisions

Collisions in which both kinetic energy and momentum are conserved.

Kinetic Energy

The energy possessed by an object due to its motion, calculated as 1/2m(v²)

Momentum Conservation in 3D

In three dimensions, momentum conservation involves summing the momentum vectors across all axes (x, y, z).

Law of conservation of energy

Energy cannot be created or destroyed, only transformed from one form to another.

Mechanical energy

The sum of kinetic and potential energies in a system.

Potential Energy (U)

The stored energy of a system due to its position or configuration.

Conservative forces

Forces that do not add energy to the system and for which the work done is independent of the path taken.

Non-conservative forces

Forces that cause energy to be transferred out of the system, like friction. Depends on the path taken.

Equilibrium points

Positions where a particle experiences no net force; can be stable, unstable, or neutral.

Stable equilibrium

A position where a slight disturbance results in a restoring force back to the initial position.

Unstable equilibrium

A position where a slight disturbance causes the particle to move further away from the initial position.

Neutral equilibrium

A position where a slight disturbance does not cause any net force or movement.

Gravitational potential energy

The potential energy associated with the gravitational position of an object, given by $U = mgy$.

Potential energy diagram

A graphical representation of potential energy as a function of position, illustrating the stability of different positions.

Work-energy principle

The principle stating that the work done on a system is equal to the change in its mechanical energy.

Energy transformation

The process of changing energy from one form to another, such as from potential to kinetic energy.

Equations of motion for conservative forces

The total mechanical energy is conserved; mathematically expressed as $U+K=0$ .

Closed Path in Work

If the work done by a conservative force around any closed path is zero, it implies returning to the initial position results in no net work.

Inelastic Collision

A collision where momentum is conserved but kinetic energy is not; some kinetic energy is transformed into other forms of energy.

Perfectly Inelastic Collision

A type of inelastic collision where the two colliding objects stick together after collision and move as one mass.

Closed System

A physical system that does not exchange matter with its surroundings and is isolated from external forces.

Momentum Conservation Equation

Initial Momentum = Final Momentum

Linear Momentum

The momentum of a particle of mass moving with velocity, defined as $p=mv$ .

Impulse

The integral of force over time. Impulse is also defined as the change in momentum of an object when a force experienced by a colliding object is applied over a period of time, expressed as J.

Newton’s Second Law

The rate of change of momentum of an object is equal to the net force applied to it. This law quantifies how external forces affect the motion of an object, stating that a net force results in acceleration proportional to the force and inversely proportional to the mass of the object.

Collision

An event in which two or more objects exert strong forces on each other for a short time.

Average Force

The force experienced by an object during a collision, which can be calculated using impulse over the time of the collision.

Work-Energy Principle

The change in potential energy associated with a conservative force is equal to the negative work done by that force.

Arbitrary Zero Reference Point

A chosen point where potential energy is defined as zero, depending on the problem context.

Negative Work

Work done by transferring energy outside of a system.

Work-Kinetic Energy Theorem

States that the net work done on a particle equals the change in the particle's kinetic energy, expressed as $W=\forall K$ .

Work (W) by a variable force.

The integral of force over distance; $W=\int Fdl.$ This measures the energy transferred when a force acts on an object causing it to move. It is calculated as the product of the force applied and the distance moved in the direction of the force.

Translational Kinetic Energy

The kinetic energy associated with the translational motion of an object, dependent on mass and speed.

Change in Kinetic Energy (∆K)

The difference in kinetic energy of an object as it moves from one state to another, expressed in the work-energy theorem.

Power

The rate at which work is done by a force.

Average Power

Calculated using the formula $P_{avg} = \frac{W}{\Delta t}$.

Instantaneous Power

Defined as $P = F \cdot v$, the product of force and velocity. Also expressed as the derivative of work with respect to time.

Units of Power

Expressed in joules per second (Js$^{-1}$) or watts (W).

Work Done by Force

When a particle moves under the influence of a constant force at an angle $\phi$, the work is calculated as $W=F\cdot d=F\cos\phi\cdot d$ .

Direction of Force

In the context of power, the force is directed at some angle $\phi$ to the x-axis.

Work (W)

The transfer of energy to an object by a force acting on it.

Line Integral

A specific form of path integral used to calculate work done by variable forces. $W=\int Fdl$.

Scalar Dot Product

The product of two vectors that results in a scalar quantity. It is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them.

Unit Vectors

Vectors that have a magnitude of one and are used to determine the direction.

Newton's Second Law of Motion

Expresses the relationship between the forces acting on an object and its acceleration, summarized as $F = ma$.

Free Body Diagram (FBD)

A graphical representation used to visualize the EXTERNAL forces acting on a body. Excludes internal forces and forces created by the body.

Friction

A force that opposes the motion of an object, which comes in static and kinetic forms.

Tension

The force exerted along a wire, rope, or cable when it is pulled tight.

Dynamic Problem Solving Strategy

A systematic approach that includes drawing diagrams, identifying forces, applying Newton’s laws, and solving equations.

Mass

The quantity of matter in an object, typically measured in kilograms (kg).

Static Friction

Frictional force acting between two systems that are in contact and stationary relative to one another.

Kinetic Friction

Frictional force acting between two systems that are in contact and moving relative to one another.

Normal Force (FN)

The force perpendicular to the contact surfaces between two objects.

Newton's three laws of motion

Fundamental principles that describe the relationship between the motion of an object and the forces acting on it.

Contact Forces

Forces that require physical contact between objects, such as normal force and friction.

Field Forces

Forces that act at a distance, such as gravitational and electromagnetic forces.

Gravitational Force

The force between particles due to their mass, which is significantly noticeable due to the Earth's mass.

Electromagnetic Forces

Forces that arise from the attraction and repulsion of charged particles and magnetic poles.

Dynamic vs Kinematic

Dynamics is the study of why objects move, while kinematics is the study of how they move.

Inertial Reference Frame

A frame of reference in which an object remains at rest or moves at a constant velocity unless acted upon by an external force where Newton’s laws of motion are valid. This doesn’t include reference frames that are accelerating or rotating, where objects appear to experience forces that cause acceleration. Inertial frames provide a consistent way to observe motion, ensuring the laws of physics hold true.