Centre of Mass: Exam-Pattern Drill (3)

hard • 2 min read
Tags Center Of Mass

Question

A man of mass 60 kg stands at one end of a 4 m long boat of mass 120 kg, floating at rest on still water. He walks to the other end. By how much does the boat shift? (Neglect water resistance.)

Solution — Step by Step

There is no external horizontal force on the man + boat system. So the centre of mass cannot move horizontally — it stays put while everything internal rearranges.

Let the boat shift by distance xx in the direction opposite to the man’s walk. Then the man’s actual displacement (in the ground frame) is 4−x4 - x in the original direction.

For COM to stay fixed:

mman(4−x)=mboat⋅xm_{man}(4 - x) = m_{boat} \cdot x

60(4−x)=120x60(4 - x) = 120 x

240−60x=120x  ⟹  x=240180=43 m240 - 60x = 120x \implies x = \frac{240}{180} = \frac{4}{3}\text{ m}

Final answer: The boat shifts by 43 m≈1.33 m\dfrac{4}{3}\text{ m} \approx 1.33\text{ m}

Why This Works

The internal force (man pushing on boat) does not move the COM. Whatever momentum the man gains, the boat gains equal and opposite. Over the walk, the boat slides backward to keep the COM at the same spot in the water.

This is the essence of “internal forces cannot move the centre of mass” — a JEE Advanced favourite.

Alternative Method

Use the formula Δxboat=mmanmman+mboat⋅L\Delta x_{boat} = \dfrac{m_{man}}{m_{man} + m_{boat}} \cdot L where LL is the length of the boat.

Δxboat=60180⋅4=43 m\Delta x_{boat} = \frac{60}{180} \cdot 4 = \frac{4}{3}\text{ m}

Same answer in one line.

Common Mistake

Students assume the man walks 4 m relative to the ground. He walks 4 m relative to the boat. In the ground frame, his displacement is less because the boat slides under his feet. Always specify which frame each displacement is measured in.

If you replace “boat” with “frictionless trolley” or “man” with “monkey on a rod”, the math is identical. JEE rewords this question every 2-3 years.

Want to master this topic?

Read the complete guide with more examples and exam tips.

Go to full topic guide →

Try These Next

Center of mass of semicircular disc of radius R

medium

Centre of Mass: Application Problems (2)

medium

Centre of Mass: Edge Cases and Subtle Traps (4)

easy

Centre of Mass: Speed-Solving Techniques (5)

medium

Centre of Mass: Step-by-Step Worked Examples (1)

easy