A tractor pulls a harrow (a ploughing tool) of mass m₁ with a net force F resulting in an acceleration of a₁. The same tractor pulls a trolley of mass m₂ with a force F producing an acceleration of a₂. If the tractor now pulls the trolley with the harrow placed on it (with the same force F), then obtain an expression for the resulting acceleration in terms of a₁ and a₂. Ignore friction.
a = (a₁a₂) / (a₁ + a₂)
Explanation
The question requires applying Newton's second law of motion to find the masses of the individual objects and then treating the combined system as a single object. According to the textbook context, Newton's second law states F = ma, and when objects are connected, they accelerate as a single system with a total mass equal to the sum of the individual masses. By finding m₁ and m₂ in terms of F, a₁, and a₂, and substituting them into the combined mass equation, we arrive at the final expression for acceleration.
Solution Steps
Step 1: Apply Newton's second law to the harrow. Given F = m₁a₁, we can express the mass of the harrow as m₁ = F / a₁.
Step 2: Apply Newton's second law to the trolley. Given F = m₂a₂, we can express the mass of the trolley as m₂ = F / a₂.
Step 3: When the harrow is placed on the trolley, they act as a single system. The total mass of the system is m₁ + m₂.
Step 4: Using Newton's second law for the combined system, the resulting acceleration is a = F / (m₁ + m₂).
Step 5: Substitute the expressions for m₁ and m₂ into the acceleration equation: a = F / (F/a₁ + F/a₂) = F / [F(1/a₁ + 1/a₂)] = 1 / (1/a₁ + 1/a₂).
Step 6: Simplify the expression to get the final answer: a = (a₁a₂) / (a₁ + a₂).