Question

A strip of wood of mass M and length l is placed on a smooth horizontal surface. An insect of mass m starts at one end of the strip and walks to the other end in time t, moving with a constant speed.

• A. The speed of the insect as seen from the ground is $<\frac{l}{t}$
• B. The speed of the strip as seen from the ground is $\frac{l}{t}\left(\frac{m}{M+m}\right)$
• C. The speed of the strip as seen from the ground is $\frac{l}{t}\left(\frac{M}{M+m}\right)$
• D. The total kinetic energy of the system is $\frac{1}{2}(m+M){\left(\frac{l}{t}\right)}^2$

Answer 1

Answer: (A), (B)

The correct options are
A The speed of the insect as seen from the ground is $<\frac{l}{t}$
B The speed of the strip as seen from the ground is $\frac{l}{t}\left(\frac{m}{M+m}\right)$


A $\to$ Centre of mass of block

B $\to$ Centre of mass of insect

Let 'C' Be centre of mass of system
where CA=x

$\Rightarrow mx=m(\frac{l}{2}-x)$

$\Rightarrow (M+m)x=\frac{ml}{2}$

$\Rightarrow x=\frac{m}{(M+m)}\frac{l}{2}-\left(i\right)$

Now, as the insect reaches the other end the displacement of centre of mass should be zero as there is no external force on the wood -insect system.

As evident from the diagram, the displacement of centre of mass of block is '2x'

From eq (i) $2x=\frac{m}{M+m}l$
Speed = $\frac{2x}{t}=\frac{l}{2}\frac{m}{(M+m)}$

Distance covered by insect from ground frame = (l-2x)

Speed of insect = $\frac{l-2x}{t}$, which is less than $\frac{l}{t}$