⚡ Work–Energy

Understanding Motion Through Work and Energy

The work–energy principle provides a direct relationship between the work done by forces and the change in the mechanical energy of a system.

Wnc = ΔE

where Wnc is the work done by non-conservative forces and E is the total mechanical energy of the system.

The animations in this section use representative problems to demonstrate how the work–energy principle is formulated and applied to systems involving gravitational potential energy, spring potential energy, and kinetic friction.

Each problem connects the physical motion, work done by forces, changes in kinetic and potential energies, and the overall energy balance.

🧮 Governing Principle

For a system moving from position 1 to position 2,

Wnc = ΔE

where, E = T + V is the total mechanical energy, consisting of the total kinetic and potential energies.

For a system with no non-conservative work, Wnc = 0, and therefore ΔE = 0 so that the total mechanical energy remains constant.

The correct application of the work–energy principle requires identification of:

• the system being analysed,

• the kinetic-energy term,

• the relevant potential-energy terms,

• the non-conservative forces doing work, and

• any kinematic constraints relating the motion of different components.

🎥 Problems Explored

1. 🌀 Collar Sliding Along a Spiral Rod

Problem: A 0.5 kg collar slides with negligible friction along a fixed spiral rod in a vertical plane under the action of a constant-magnitude radial force T = 10 N. The rod is described by r(θ) = 0.3 θ, with the collar moving from A to B.

Concept: Application of the work–energy principle to curvilinear motion along a prescribed spiral path, with the work done by a constant radial force evaluated from the change in radial position.

🔍 Watch For:

• How the collar follows the prescribed spiral path r(θ) = 0.3 θ.

• How the motion involves both radial and transverse components of velocity.

• How the constant radial force T does work as the radial coordinate changes.

• How the work done by the radial force is evaluated from Wₜ = ∫ T · dr

• How gravitational potential energy changes as the collar moves through the vertical plane.

• How the reaction force from the fixed rod does no work because it is perpendicular to the instantaneous displacement.

• How the kinetic energy changes during the motion.

• How the total mechanical energy increases as a result of the positive work done by the applied radial force.
  
  

Observe the live verification showing that the work done by the non-conservative force is equal to the change in total mechanical energy:

ΔE − ∫T · dr = 0 

2. 🛝  Block–Pulley–Spring System with Friction

Problem: A 10 kg block moves down a rough 50° incline with an initial speed of 0.3 m/s. The block is connected through a cable and pulley system to a spring of stiffness k = 200 N/m, initially stretched by 25 mm. The coefficient of kinetic friction is μₖ = 0.15.

Concept: Application of the work–energy principle to a constrained system involving gravitational potential energy, spring potential energy, and work done by kinetic friction.

🔍 Watch For:

• How the cable constraint relates the displacement of the block to the displacement of the spring end.

• How the spring displacement is twice the block displacement, δs = 2s.

• How the initial spring extension contributes to the initial spring potential energy.

• How gravitational potential energy changes as the block moves down the incline.

• How spring potential energy changes as the spring is stretched.

• How the kinetic energy changes as the block moves.

• How kinetic friction continuously removes mechanical energy from the system.

• How the normal reaction does no work because it is perpendicular to the block's displacement.

• How the block eventually comes to rest when its kinetic energy becomes zero.

• How the work–energy balance can be used to determine both the velocity after a specified displacement and the distance travelled before the block comes to rest.

Observe the live verification showing:

ΔE − Wfk  = 0

where ΔE is the change in total mechanical energy and Wfk  is the work done by kinetic friction.

💡 Key Idea

The work–energy principle provides a direct relationship between the work done by non-conservative forces and the change in total mechanical energy:

Wnc = ΔE

The principle is particularly useful when the desired quantity is related to velocity or displacement, and determining the complete acceleration history is unnecessary.

The key steps are:

1. Identify the system and its motion.

2. Identify the kinetic and potential-energy terms.

3. Identify the non-conservative forces doing work.

4. Apply the energy balance between the initial and final states.

5. Use any kinematic constraints required to relate the motion of different components.