This chapter covers the concepts of work, energy, and power in physics, exploring the relationships between them, the work-energy theorem, and the conservation of mechanical energy.
Work, energy, and power are commonly used terms in everyday language, but their definitions in physics are precise.
Work refers to the energy transferred when a force is applied to an object.
Energy signifies the capacity to do work; it reflects the ability to cause change.
Power indicates the rate at which work is done or energy is transferred.
5.2 NOTIONS OF WORK AND KINETIC ENERGY: THE WORK-ENERGY THEOREM
The relationship defined by the work-energy theorem states that the change in kinetic energy of an object is equal to the work done on it by the net force.
Mathematically:
$$ K_f - K_i = W_{net} $$
where:
$K_f$ is the final kinetic energy,
$K_i$ is the initial kinetic energy,
$W_{net}$ is the work done on the object.
WORK DONE BY A VARIABLE FORCE
For constant and varying forces, the work done can be expressed as:
$$ W = ext{Force} imes ext{Displacement} imes ext{cos}( heta) $$
where $ heta $ is the angle between the force and the displacement vectors.
In terms of energy, this means work can change the kinetic energy of a body:
$$ W = rac{1}{2} mv^2 - rac{1}{2} mu^2 $$
5.3 WORK
The definition of work in physics emphasizes displacement; no displacement means no work done despite exerting force.
Common situations include:
Lifting an object (positive work),
Frictional force when sliding down an incline (negative work).
5.4 KINETIC ENERGY
The kinetic energy (K) of an object is:
$$ K = rac{1}{2} mv^2 $$
where m is mass and v is velocity.
This reflects the object's capacity to perform work due to its motion.
5.5 WORK DONE BY A VARIABLE FORCE
Work done by a variable force can be calculated by integrating the force over the path taken:
$$ W = extstyle igintss_{x_i}^{x_f} F(x) extrm{d}x $$
5.6 THE WORK-ENERGY THEOREM FOR A VARIABLE FORCE
For variable forces, the work-energy theorem still applies:
$$ K_f - K_i = W $$
The integration helps determine the work based on varying forces acting along a displacement.
5.7 THE CONCEPT OF POTENTIAL ENERGY
Potential energy (V) represents stored energy based on an object's position:
For gravitational systems:
$$ U = mgh $$
where h is height above a reference level.
In simple harmonic motion,
$$ V(x) = rac{1}{2} kx^2 $$
where k is spring constant and x is displacement from equilibrium.
5.8 CONSERVATION OF MECHANICAL ENERGY
For a closed system with only conservative forces acting, total mechanical energy is conserved:
$$ E = K + V = ext{constant} $$.
Changes in kinetic energy are compensated by changes in potential energy.
5.9 THE POTENTIAL ENERGY OF A SPRING
Hooke’s Law defines spring force as:
$$ F = -kx $$.
The potential energy stored in a spring when compressed or extended is:
$$ V(x) = rac{1}{2} kx^2 $$.
5.10 POWER
Power (P) is the rate at which work is done:
$$ P = rac{W}{t} $$
Its unity is the watt (W), defined as 1 Joule per second (1 W = 1 J/s).
5.11 COLLISIONS
In collisions:
Momentum is conserved.
Kinetic energy is conserved in elastic collisions but not in inelastic collisions.
Mathematically, for elastic collisions:
$$ m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} $$
and the kinetic energy is given:
$$ rac{1}{2} m_1{v_{1i}}^2 + rac{1}{2} m_2{v_{2i}}^2 = rac{1}{2} m_1{v_{1f}}^2 + rac{1}{2} m_2{v_{2f}}^2 $$.
KEY POINTS
Work is defined as the product of force and displacement in the direction of the force.
The work-energy theorem connects work done to changes in kinetic energy.
Kinetic energy is calculated as ( K = \frac{1}{2} mv^2 ).
Potential energy is defined for gravitational and elastic systems.
The conservation of mechanical energy principle states total energy remains constant if only conservative forces act.
The definitions of power relate work done over time; it has units of watts.
During collisions, momentum is conserved; the conservation of kinetic energy depends on the type of collision (elastic vs. inelastic).
The scalar product of two vectors gives a scalar and assists in computing work.
Forces can be categorized as conservative (storable energy) or non-conservative (like friction, no stored energy).
Key terms/Concepts
Work is the energy transferred by a force acting through a distance.
The work-energy theorem relates work done to a change in kinetic energy.
Kinetic energy is given by ( K = \frac{1}{2} mv^2 ), indicating the work required to change motion.
Potential energy signifies stored energy due to position, including gravitational and elastic potential energy.
The conservation of mechanical energy states that the total energy in a closed system remains constant.
Power measures the rate of doing work or transferring energy, measured in watts (W).
In collisions, momentum is always conserved, while kinetic energy conservation depends on whether the collision is elastic or inelastic.
Scalar product provides a method to compute work, acting as the product of magnitudes and the cosine of the angle between vectors.