Connected Bodies (Lifts)

Newton's Second Law ($F = ma$) is the primary tool for solving lift problems. It states that the resultant force in the direction of motion is equal to the mass of the object multiplied by its acceleration.

Newton's Third Law explains the interaction between the lift floor and the load. If the floor pushes up on the person with force $R$, the person pushes down on the floor with the same force $R$.

Weight ($W = mg$) is a constant force acting vertically downwards. In lift problems, we must distinguish between the weight of the lift ($Mg$) and the weight of the load ($mg$).

Method 1: Whole System Analysis

• Treat the lift and the load as a single particle with a combined mass of $(M + m)$.
• In this view, the internal reaction force $R$ cancels out because it acts in opposite directions on the two components.
• The equation of motion becomes $T - (M + m)g = (M + m)a$ (assuming upward acceleration).

Method 2: Separate Component Analysis

• Analyze the load only: The forces are the reaction $R$ (up) and its weight $mg$ (down). The equation is $R - mg = ma$.
• Analyze the lift only: The forces are tension $T$ (up), its weight $Mg$ (down), and the reaction $R$ from the load (down). The equation is $T - Mg - R = Ma$.

Decision Criteria

• Use the Whole System method when you need to find the tension $T$ or the acceleration $a$ without needing the internal reaction force.
• Use the Separate Component method when the question specifically asks for the reaction force $R$ or the mass of the load.

It is vital to distinguish between the Tension in the cable and the Reaction Force on the floor. Tension supports the entire mass, while the reaction force only supports the load resting on the floor.

The direction of acceleration determines the sign of the forces. If the lift is accelerating upwards, the upward forces ($T$ or $R$) must be greater than the downward forces (weight).

• Define a Positive Direction: Always start by choosing a positive direction (usually the direction of acceleration) and stick to it for all equations in that problem.

• Check the Motion State: If the lift is moving at a constant velocity, the acceleration $a = 0$. This means the system is in equilibrium, and $T = (M+m)g$ and $R = mg$.

• Rounding and $g$: If using $g = 9.8 \text{ m/s}^2$, ensure your final answer is rounded to 2 significant figures. If using $g = 10 \text{ m/s}^2$, round to 1 significant figure.

• Hidden Lifts: Be prepared for 'hidden' lift scenarios, such as a crate being raised by a crane or a fairground ride. The physics remains identical to a standard elevator problem.