Connected Bodies (Lifts)

• Newton's Second Law ($F = ma$) is the primary tool for solving these problems. The resultant force in the direction of motion is equated to the product of the mass and acceleration.

• Newton's Third Law (N3L) dictates that the force the lift floor exerts on the load ($R$) is equal in magnitude and opposite in direction to the force the load exerts on the lift floor.

• Gravity ($W = mg$) acts vertically downwards on every component of the system. In lift problems, the acceleration due to gravity ($g$) is typically taken as $9.8 \text{ m s}^{-2}$.

Method 1: Treating as a Single System

• Application: Use this when you need to find the Tension ($T$) in the cable and you know the total mass ($M + m$).
• Equation: For upward acceleration, the equation is $T - (M + m)g = (M + m)a$. This ignores internal reaction forces because they cancel out as an action-reaction pair.

Method 2: Treating Particles Separately

• Application: Use this when you need to find the Reaction Force ($R$) between the floor and the load.

• Load Equation: For the load alone, the equation is $R - mg = ma$ (if accelerating upwards). This isolates the load from the lift's mass.

• Lift Equation: For the lift alone, the equation is $T - R - Mg = Ma$. Note that $R$ acts downwards on the lift floor while acting upwards on the load.

• Constant Velocity vs. Acceleration: If the lift moves at a constant velocity, the acceleration $a = 0$. In this state, $R = mg$ and $T = (M + m)g$, meaning the forces are in equilibrium.

• Upward vs. Downward Acceleration: Direction matters for the sign of $a$. If the lift is accelerating downwards, the weight exceeds the upward forces, leading to equations like $mg - R = ma$.

• Define a Positive Direction: Always start by choosing whether 'up' or 'down' is positive and stick to it for all equations in that problem. Usually, choosing the direction of acceleration as positive simplifies the algebra.

• The 'Hidden Lift': Be prepared for scenarios that don't use the word 'lift,' such as a crate being lowered by a crane or a person on a scale in a moving vehicle. The physics remains identical.

• Sanity Check: If a lift is accelerating upwards, $R$ must be greater than $mg$. If your calculated $R$ is smaller than the weight while accelerating up, you likely have a sign error.

• Units and Mass: Ensure all masses are in kilograms (kg). If a problem gives mass in tonnes, convert it immediately ($1 \text{ tonne} = 1000 \text{ kg}$).