Connected Bodies (Lifts)

• Newton's Second Law ($F = ma$): The resultant force acting on any part of the system (or the whole system) is equal to the mass of that part multiplied by its acceleration.

• Newton's Third Law (N3L): The force the floor exerts on the load (Normal Reaction, $R$) is equal in magnitude and opposite in direction to the force the load exerts on the floor.

• Gravity ($g$): Weight is calculated as $W = mg$, where $g$ is the acceleration due to gravity (approximately $9.8 \text{ m s}^{-2}$).

• Resultant Force Calculation: To find the resultant force, you must subtract the sum of forces acting in the negative direction from the sum of forces acting in the positive direction.

Method 1: Treating the System as One

• Application: Use this when you need to find the acceleration ($a$) or the tension ($T$) in the main cable without needing the internal reaction force ($R$).

• Equation: $(T - (M + m)g) = (M + m)a$ (assuming upwards is positive), where $M$ is the mass of the lift and $m$ is the mass of the load.

Method 2: Treating the Load Separately

• Application: Use this to find the normal reaction force ($R$) between the floor and the load.

• Equation: $(R - mg) = ma$ (assuming upwards is positive).

Method 3: Treating the Lift Separately

• Application: Use this if you need to isolate the forces acting specifically on the lift cage, such as when the tension and reaction are both unknown.

• Equation: $(T - R - Mg) = Ma$ (assuming upwards is positive).

• Tension vs. Reaction: Tension is the force in the external cable pulling the entire lift, while Reaction is the internal contact force between the lift floor and the object inside.

• Define a Positive Direction: Always explicitly state which direction is positive (usually the direction of acceleration) to ensure consistent signs in your equations.

• Check the Units: Ensure all masses are in kilograms (kg) and forces are in Newtons (N). Convert tonnes to kg ($1 \text{ tonne} = 1000 \text{ kg}$) immediately.

• Hidden Lifts: Be prepared to apply these principles to non-standard scenarios like a crate being lifted by a crane or a person on a fairground drop tower.

• Sanity Check: If a lift is accelerating upwards, the reaction force $R$ should be greater than the weight $mg$. If it is accelerating downwards, $R$ should be less than $mg$.

• Forgetting Gravity: A common error is using mass ($m$) instead of weight ($mg$) in the force side of the $F=ma$ equation.

• Mixing Masses: Students often use the total mass ($M+m$) when they should only be using the load mass ($m$) to calculate the reaction force $R$.

• Sign Errors: Incorrectly assigning signs to forces (e.g., making both Tension and Weight positive) will lead to an incorrect resultant force.

• Constant Velocity: Remember that if a lift is moving at a constant velocity, the acceleration $a = 0$, meaning the resultant force is zero ($T = W$ and $R = mg$).