WORK, ENERGY AND MOMENTUM

1. Work and Kinetic Energy

Work and Kinetic Energy

Work and kinetic energy are connected by the work-energy theorem: the net work done on an object equals the change in its kinetic energy.

Work is done when a force moves an object through a distance in the direction of the force. If the work done on an object is positive, its kinetic energy increases; if the work is negative, its kinetic energy decreases.

The relationship is written as:

where   is mass,   is initial velocity, and   is final velocity.

This means work transfers energy to or from an object. For example, when you push a cart and it speeds up, the work you do increases its kinetic energy.

Example

If a stationary ball is kicked and begins moving, the kick does work on the ball, and that work becomes kinetic energy. The greater the force or the distance moved, the more work is done and the more the kinetic energy changes.

Definition: work is defined as the product of Force and distance in the direction of the applied force. It's is simply the application of Force to move an object over a distance d, in the direction of the applied force. In addition, it's described by the equation: W = f.d

Example.

1. Doing homework. It's not work because objects are not moved from one place to another over a distance

2. Lifting a rock upwards off the ground. It work-done because the rock is moving toward the applied force.

3. Carrying a rock in a straight line or path across a lawn at a constant speed. It's not working. Recall from the laws of motion that force is not required to move an object at constant velocity.

Work done W = fxs

Or

W = mgxs                                        

Resolving work in terms of magnitude and direction.

Let's assume force is applied at an angle to the horizontal or vertical. The work done will depend on the direction the object moves in.

Assume the box moves in the horizontal direction

W = Fcosxs or W = mgcosxs

Again, assume the box moves in the vertical direction, W = Fsinxs or mgsinxs

Where F = force in Newtons

M = mass in kg

S = distance in meters

g = acceleration due to gravity in ms^-2

w = work in Joules

Kinetic Energy:  It is the energy an object has because of its motion. The faster an object moves, or the more mass it has, the greater its kinetic energy. Any moving object has kinetic energy, whether it is a rolling ball, a moving car, or flowing water. If the object stops moving, its kinetic energy becomes zero. It depends on the mass of an object and its velocity

Energy is defined ability or capacity to do work. Energy can take a variety of forms, and one form of energy can transform into another.

Potential energy: Also called stored energy, comes in several forms.  Potential energy is the energy stored in an object because of its position, condition, or arrangement. In simple terms, it is energy that has the potential to be converted into motion or other forms of energy later.  For instance, if a force is applied to lift a rock off the ground, it increases the rock's potential energy, PE. However, if we drop the rock, the force of gravity increases the rock's KE as the rock moves downward until it hits the ground. The force exerted to lift the rock is equal to its weight, which is equal to its mass, multiplied by acceleration due to gravity g and the distance to which the rock was lifted F = W = mg. Hence PE = mgh (Energy possessed by a body at rest or height)

Examples:

1. A constant force of 80N acting on a body initially at rest gives an acceleration of 0.3 m/s2 for 8s. Calculate the work done by the force.

2. A boy drags a bag of Rice along a smooth horizontal floor surface with a force of 2N applied at a. Angle 60° to the floor. Determine the work done at distance of 3m.

3. A pulley of efficiency lifts 54kg of water through a height of 33m in 12s. Calculate the power of the pulley?

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in an object due to its position in a gravitational field, usually because it is raised above some reference level (such as the ground). This energy can be converted into kinetic energy when the object falls under gravity, obeying the law of conservation of energy.

Gravitational potential energy is a type of potential energy associated with the gravitational force between masses. For an object near Earth’s surface, GPE is the work done to lift the object against gravity from a chosen zero‑level (often the ground) to its present height, without changing its speed. Example:

·         When you lift a book and hold it above the desk, the chemical energy in your muscles is partly stored as gravitational potential energy in the book–Earth system.

For most introductory physics problems, where the gravitational field is approximately constant (magnitude), the gravitational potential energy of a mass   at height   above a reference level is U=mgh

Note:

• The value of   depends on the choice of the reference level (you can set   at the ground, floor, tabletop, etc.), but changes in GPE are what matter for energy calculations.
• GPE increases with mass, height, and the strength of the gravitational field.

2. Relationship between work and energy conservation

Relationship between work and energy conservation

Gravitational potential energy is defined via work done by or against gravity:

• The change in GPE is equal to the negative of the work done by the gravitational force:
• When only conservative forces (like gravity) act, total mechanical energy (kinetic + potential) is conserved:

For instance, a ball dropped from height   has Initial: high, zero   and at impact:  ,  .

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. In a closed (or isolated) system, the total amount of energy remains constant over time.

Energy may change from forms such as kinetic, gravitational potential, elastic, thermal, chemical, or electrical, but the sum of all energies in the system stays fixed, provided no energy enters or leaves the system.
If energy does enter or leave (for example, via heat or work), then the change in the system’s total energy equals that energy transfer, as described in the first law of thermodynamics.

Forms of energy

1. Kinetic energy (energy of motion).
2. Gravitational potential energy 
3. Elastic potential energy in springs.
4. Thermal energy (heat), internal energy, and other forms in extended systems.

Conservation of mechanical energy

In many A‑level style problems, only mechanical energy (kinetic plus potential) is considered.
When all forces are conservative (no friction or drag), mechanical energy is conserved:

where subscripts   and   stand for initial and final states.

Example:

• A pendulum bob swings down: gravitational potential energy converts into kinetic energy, with total mechanical energy roughly constant if air resistance is small.

Non‑conservative forces and real systems

In real systems, non‑conservative forces like friction or air resistance convert mechanical energy into heat, sound, etc.
The total energy of the larger system (including the surroundings) still obeys conservation of energy, but mechanical energy alone decreases.

Linear and Angular momentum

Linear momentum and angular momentum are both measures of “how much motion” an object has, but they apply to different kinds of motion: translation (straight‑line) and rotation. Both obey conservation laws when there are no external forces or torques, and they are central in mechanics‑style problems.

Linear (translational) momentum

Linear momentum   of a particle is defined as:

where   is mass and   is its linear velocity.

• It is a vector quantity; its direction is the same as the velocity.
• For a system of particles, the total linear momentum is the vector sum:  .

Conservation of linear momentum
If the net external force on a system is zero, its total linear momentum remains constant

This is why momentum is very useful in collision problems (e.g., car crashes, particle collisions).

Angular momentum

Angular momentum   describes rotational or “orbital” motion about a point or axis.

For a single particle

For a particle of mass   at position   relative to an origin and moving with linear momentum  , the angular momentum about that origin is
In magnitude, for a case where   and   are perpendicular (e.g., circular orbit), this becomes

For a rigid body rotating about an axis

For a rigid body spinning with angular velocity, its angular momentum is
where   is the moment of inertia about the axis and   is the angular speed.

Conservation of angular momentum

If the net external torque on a system about a given axis is zero, its total angular momentum about that axis is conserved:

Consequences:

• A spinning ice skater pulling arms in reduces, so   increases to keep   constant.
• Planets in orbit have constant angular momentum (no external torque), which leads to Kepler’s second law (equal areas swept in equal times).

How linear and angular momentum are linked

Angular momentum can be thought of as the “rotational cousin” of linear momentum: replace mass with moment of inertia and linear velocity with angular velocity.
Many rotational formulas are direct analogues of their translational counterparts; for example,

3. System of particles and center of mass

System of particles and center of mass

A system of particles is a collection of point‑mass objects whose motion you analyze together (for example, colliding bodies, fragments of an explosion, or planets in a cluster). The center of mass (COM) of such a system is a special point that represents the “average” position of all the mass and behaves like a single particle whose mass equals the total mass of the system.

Definition of center of mass

Center of mass (COM) is the point at which the entire mass of a system may be considered to be concentrated for analysing its motion. It is the point where the system balances if supported.

For a system of   particles with masses   at positions, the position vector of the center of mass is:

where   is the total mass of the system.
In coordinates, this becomes:

Note:

The center of mass is not the geometric center unless all masses are equal and symmetrically placed.

It is “weighted” by the masses: a heavier particle pulls the COM closer to itself.

The COM point moves as if all the mass of the system were concentrated there, and all external forces acted on that point.

• The total linear momentum of the system equals the momentum of a single particle of mass   moving with the velocity of the COM:  .
• For a body with symmetry (e.g., uniform sphere, cylinder, rectangular block), the COM lies at the geometric center or on the axis of symmetry.

Properties of center of mass

        i.            It is the balance point of a body or system

      ii.            It may lie inside or outside the body

    iii.            If no external force acts on the system, the center of mass moves with constant velocity.

Examples:

Two masses 2kg and 3kg are placed at positions 0 and 4m respectively. Determine the center of mass.

Three particles have masses 2kg, 5kg and 7 kg are located at 2m, 4m and 8m respectively. Find it center of mas.

Three particles are located at 2kg at (2,3), 3kg at (4, 5) and 5kg at (6,1). Find the coordinates of the center of mass.

A 4kg mass is 2m and unknown mass is 8m. If the center of mass is at 5m, find the unknown mass.

 Motion of the center of mass

The motion of the center of mass (COM) describes how the average position of all the mass in a system change with time. Regardless of the interna motions of the particles, the center of mass moves as if all the mass of the system were concentrated at one point and all the external forces acted at that point.

If the net external force on the system is, then Newton’s second law for the COM is:

Where Fxt = external force

M = total mass of the system

a_cm = acceleration of the center of mass

This implies that Internal forces (between particles) do not affect the COM motion; they cancel in pairs by Newton’s third law.

Explosions, collisions, and internal rearrangements change the shapes and velocities of the parts, but the path of the COM is still governed by   alone.

Example:

1. A system has a total mass of 20kg. An external force of 100N acts on it . Find the acceleration of the center of mass.