Work Done by Constant Force

Ever wonder how physicists measure the effort it takes to move something? That's where work comes in! Work happens when you apply a force that causes an object to move.

Work depends on two critical factors: the force applied and the displacement of the object. When a force acts in the same direction as movement, work is calculated using a simple formula:

W = F × S

What if the force isn't directly aligned with the movement? Then we need to consider only the component of force in the direction of motion. The formula becomes:

W = F × S × cos θ

Remember This! Work is a scalar quantity (it has magnitude but no direction) and equals the dot product of force and displacement vectors. The angle between force and displacement determines whether work is positive, negative, or zero.

Special Cases of Work

The direction of force relative to movement creates three interesting situations for work.

Positive Work occurs when force and displacement point in the same direction $θ = 0°$. This happens when you push an object forward or lift something upward. The formula simplifies to W = F × S, making calculations straightforward.

Zero Work happens when force and displacement are perpendicular to each other $θ = 90°$. A weight lifter walking horizontally while holding weights does zero work on the weights! Similarly, centripetal forces and the tension in a pendulum string perform zero work.

Negative Work occurs when the force opposes the direction of displacement $θ = 180°$. When you lower a bucket into a well, your upward pulling force does negative work. Friction always performs negative work because it opposes motion.

The standard unit of work is the joule (J), named after James Prescott Joule. One joule equals the work done when a 1 newton force moves an object 1 meter in the direction of the force.

Units of Work and Variable Forces

Work can be measured in different units depending on the system:

In the CGS system, work is measured in ergs (1 erg = 1 dyne-cm). The British Engineering system uses foot-pounds. For atomic physics, we use electron-volts (eV), where 1 eV = 1.6 × 10⁻¹⁹ J.

But what happens when force isn't constant? For a variable force in one dimension, we divide the total displacement into tiny intervals where the force is approximately constant. To find the total work, we sum all these small contributions:

W = ∫ F$x$ dx

This integration from initial position xᵢ to final position xf allows us to calculate work for any variable force. For example, when a spring stretches or compresses, the force constantly changes as the spring moves.

Physics Insight: When dealing with variable forces, calculus becomes our best friend! Breaking complex movements into infinitesimally small pieces allows us to add up work contributions accurately.

Work Done by Variable Forces

When a force varies as an object moves, the calculation gets more interesting. Let's look at how we handle this mathematically.

For a variable force, we break the total displacement into tiny segments (δx) where the force is approximately constant. The total work becomes:

W = ∑ F̄ₙ × δxₙ

As we make these segments infinitesimally small (δx → 0), this sum becomes an integral:

W = ∫ F$x$ dx

A perfect example is a spring-mass system, where force changes with position. When you stretch or compress a spring, it pushes back with greater force the more you displace it. This relationship follows Hooke's Law, which states that the restoring force is proportional to displacement:

F = -kx

The negative sign indicates the force opposes the displacement. The constant k tells you how stiff the spring is - higher values mean stiffer springs that require more force to stretch.

Try This: Next time you play with a spring, notice how the force needed increases as you stretch it further. This is Hooke's Law in action!

Spring Forces and Work

The spring is a perfect example of a variable force system. When you stretch or compress a spring, it pushes back with a force that changes with position.

The restoring force of a spring follows Hooke's Law: F = -kx

Where:

• k is the spring constant (higher values mean stiffer springs)
• x is displacement from equilibrium
• The negative sign shows the force opposes displacement

When calculating work done by a spring as it moves from position xᵢ to xf:

Wₛ = ∫ $-kx$ dx = -½k(xf² - xᵢ²)

This means when a spring moves from a stretched position toward its natural position (xᵢ > xf), it does positive work on the attached object. Conversely, when stretching a spring further (xᵢ < xf), the spring does negative work.

For a spring initially at rest $xᵢ = 0$ that's stretched to position x: Wₛ = -½kx²

Cool Connection: The work required to stretch a spring equals the energy stored in it. This is why springs are such useful energy storage devices in everything from mattresses to mechanical watches!

Work in Two Dimensions

Real-world forces often vary in both magnitude and direction, with objects moving along curved paths. How do we calculate work in these cases?

When a force F⃗ varies along a curved path from point i to point f, we:

1. Break the path into tiny displacement vectors δs⃗
2. Calculate work for each segment: δW = F⃗·δs⃗ = F cos φ δs
3. Add all these contributions using integration

The total work becomes: W = ∫ F⃗·ds⃗ = ∫ F cos φ ds

This is called a line integral because we're integrating along a path. To evaluate it, we express both force and displacement in terms of their components:

F⃗ = Fₓî + Fyĵ ds⃗ = dxî + dyĵ

The work integral becomes: W = ∫(Fₓdx + Fydy)

For three-dimensional motion, we simply add the z-component: W = ∫(Fₓdx + Fydy + Fzdz)

Practical Application: This is how engineers calculate energy requirements for objects following complex paths, like roller coasters or satellites in orbit!

Kinetic Energy and Work-Energy Theorem

One of physics' most powerful ideas connects work and energy: The Work-Energy Theorem states that the net work done on a particle equals the change in its kinetic energy.

When a net force acts on an object and moves it from position xᵢ to xf, causing velocity to change from vᵢ to vf, the work done is:

Wnet = ½m(vf² - vᵢ²) = Kf - Kᵢ = ΔK

Where K = ½mv² is the object's kinetic energy - the energy it possesses due to motion.

This remarkable result means whenever work is done on an object, its kinetic energy changes by exactly that amount. The work-energy theorem holds for both constant and variable forces!

For a formal proof, we start with: Wnet = ∫Fnetdx

Since Fnet = ma = m(dv/dt), and with some calculus manipulations: Wnet = ∫mv dv = ½m(vf² - vᵢ²)

Why This Matters: The work-energy theorem gives us an alternative to Newton's second law for solving many physics problems. Instead of tracking forces and accelerations, we can focus on work and energy!

Power: The Rate of Doing Work

How quickly can you perform work? That's what power measures - the rate at which work is done.

If a force F⃗ does work dW in a small time interval dt:

P = dW/dt

Since dW = F⃗·ds⃗, and velocity v⃗ = ds⃗/dt, power can be written as:

P = F⃗·v⃗

This elegant formula shows power equals the dot product of force and velocity vectors. It reveals that applying the same force will generate more power if you move faster!

There are two important ways to express power:

Average Power over time interval t: P̄ = W/t

Instantaneous Power at a specific moment: P = dW/dt

The standard unit of power is the watt (W), named after James Watt. One watt equals one joule per second $1 W = 1 J/s$.

Other power units include:

• Erg per second (CGS system)
• Horsepower (1 HP = 746 watts)

Everyday Example: A 100-watt light bulb consumes energy at the rate of 100 joules every second. This is why higher wattage appliances cost more to run!

Power and Work Relations

Power and work are closely connected concepts, with power telling us how quickly work is accomplished.

When power stays constant, the relationship becomes simple: W = P × t

This is why energy consumption is often measured in kilowatt-hours (kWh) - a unit that combines power (kilowatts) and time (hours) to give total energy used.

Let's see how power relates to force and velocity in a practical situation. Consider pushing a sled across snow:

P = F⃗·v⃗

If force and velocity point in the same direction, this simplifies to: P = F × v

This means you can increase power by either:

• Applying more force, or
• Moving faster

For variable power situations, we use the instantaneous formula: P = dW/dt

The SI unit of power is the watt $1 W = 1 J/s$, but other units include:

• Horsepower: 1 HP = 746 W = 550 ft-lb/s (commonly used for engines)
• Kilowatt-hour: A unit of energy equal to 3.6 million joules

Real-World Connection: When a car manufacturer lists horsepower, they're telling you the maximum rate at which the engine can perform work. Higher horsepower means the car can accelerate faster or climb hills more easily!

Reference Frames

When analyzing motion and work, where you make your measurements matters. A reference frame is the coordinate system in which measurements are made.

There are two main types of reference frames:

Inertial reference frames are either at rest or moving with constant velocity. Newton's laws of motion work perfectly in these frames. While no absolutely perfect inertial frame exists in nature, many situations can be approximated as inertial. For example:

• A coordinate system fixed to Earth is approximately inertial for most everyday situations
• The Earth rotates and orbits the sun, creating small accelerations (about 3.4 cm/s² at the equator)
• The sun orbits the galaxy with tiny acceleration (about 3×10⁻⁸ cm/s²)

Non-inertial reference frames are accelerating relative to inertial frames. Newton's first law doesn't hold in these frames without modification. Common examples include:

• A car braking suddenly
• A rotating merry-go-round
• An elevator accelerating upward

Think About It: Have you noticed how you feel pushed backward when a car accelerates? That's because you're in a non-inertial reference frame where Newton's laws need adjustment to explain what you feel!