Work and Force-Displacement Graphs

Work

• Represented by the symbol W
• Constant force on an object is the product of the force component along the displacement direction, and the displacement magnitude.
• W = (F cos θ)∆x
• F is the magnitude of the force
• Δx is the magnitude of the object’s displacement
• θ is the angle between F and Δx
• Scalar quantity
• The S.I. unit of work is kg m² s⁻² or Joule (J)
• 1 Joule is the work done by a 1 N force that results in a 1 m displacement in the force's direction
• No work occurs if there is no movement
• No work occurs if the force is perpendicular to the displacement

Force-Displacement Graphs

• Illustrates how the force acting on an object changes as the object moves
• The movement is due to displacement in the force's direction.
• It helps calculate work done by the force
• To do this, find the area under the graph.
• Consists of displacement, s (m) on the x-axis
• Consists of force, F (N) on the y-axis

Work Can Be

• Positive if the force has a component in the motion direction
• Zero if the force has no component in the motion direction
• Negative if the force has a component opposite the motion direction

Positive Work

• A force F applied in the moving car's direction, accelerates it
• The force and displacement are in the same direction where θ = 0°
• The work done is given by W = Fx cos 0° = Fx
• The work is positive

Negative Work

• A force F applied in opposite direction of the car, brings it to rest after some distance
• The force F is applied in the opposite direction to stop the car, where θ = 180°
• Work done is given by W = Fd cos 180° = –Fd
• The work done by the force is negative

Energy

• Energy is a system's ability to do work
• The S.I. unit for energy is Joule, J
• Energy is a scalar quantity

Kinetic Energy

• It is the energy of a body due to its motion
• Kinetic Energy = ½mv²
• m = Mass (in kg)
• v = Velocity (in m/s)

Potential Energy

• The energy stored in a body/system because of position, shape, and state
• Gravitational potential energy is the energy stored in a body/system because of its position
• Potential Energy (GPE) = mgh
• m = Mass (in kg)
• g = Acceleration of gravity (9.8 m/s²)
• h = Height (in meters)

Elastic Potential Energy U

• Energy stored in elastic materials when stretched/compressed
• Springs are a special instance of device which can store elastic potential energy due to its compression or stretching.
• Hooke’s Law states that the restoring force, F of a spring is directly proportional to the amount of stretch/compression (extension), x if the proportionality limit isn't exceeded.
• A constant k is called the spring constant/force constant and is measured in N/m
• The spring constant is the slope of the line in a force-extension graph.
• An ideal spring obeys Hooke’s Law within compression/stretching limits, beyond which it may deform.

Conservation of Energy

• Energy cannot be created nor destroyed
• Energy can only be converted from one form to another
• Mechanical Energy is what objects have when in motion and/or at some position

Total Mechanical Energy

• Total Mechanical Energy is the sum of kinetic energy and all forms of potential energy
• Kinetic Energy (Energy of motion) is expressed as: KE = ½ mv²
• Potential Energy is the stored energy of position
• Gravitational Potential Energy = PEg = mgh
• Elastic Potential Energy = PEelastic = ½ kx²

Conservation of Energy

• In the absence of friction, mechanical energy is conserved, so the amount of mechanical energy is constant
• Initial(i) mechanical energy = final (f) mechanical energy
• PEi + KEi = PEf + KEf, expanded to mghi + ½ mvi² = mghf + ½ mvf²

Work-Kinetic Energy Theorem

• Net work done equals the change in kinetic energy (KE)
• W = ΔKE
• W = KEf − KEo = ½ mvf² − ½ mvo²

Power

• The rate at which work is done or the rate at which energy is transferred
• Scalar quantity measured in Watts (W)
• If an amount of work, W is done during an amount of time ∆t by a force
• The equation for Average Power is: Pav = ΔW/Δt = ΔE/Δt

Mechanical Efficiency

• It measures the performance of machines/engines
• It is the ratio of the useful (output) work done to the energy input
• No unit of measure