Embibe Experts Solutions for Chapter: Work, Energy and Power, Exercise 4: Exercise - 4

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $\rho$ as shown in the figure. The height of the liquid in one vessel is $h_2$ and in other vessels $h_1$, the area of either base is $A$. Find the work done by gravity in equalizing the levels when the two vessels are connected.

A uniform rope of linear mass density $\mathrm{\lambda }$ and length $l$ is coiled on a smooth horizontal surface. One end is pulled up with constant velocity $\mathrm{v}.$ Then find average power applied by the external agent in pulling the entire rope just off the ground ?

A uniform rope of linear mass density $\lambda$ and length $l$ is coiled on a smooth horizontal surface. One end is pulled up with a constant velocity $v$. The maximum power delivered by the agent in pulling up the rope is?

Two blocks $A \& B$ of mass $m \& 2m$ respectively are joined to the ends of an undeformed massless spring of spring constant $k$. They can move on a horizontal smooth surface. Initially $A \& B$ have velocities $u$ towards left and $2u$ towards right respectively. Constant forces of magnitudes $F \text{and} 2F$ are always acting on $A \text{and} B$ respectively in the directions shown. Find the maximum extension in the spring during the motion.

A chain of mass $M$ and length $L$ is held vertically such that its bottom end just touches the surface of a horizontal table. The chain is released from rest. Assume that the portion of chain on the table does not form a heap, the momentum of the portion of the chain above the table after the top end of the chain falls down by a distance $\frac{L}{8}$,

From what minimum height $h$ in metre must the system be released when the spring is in its natural length as shown in the figure so that after a perfectly inelastic collision $(e=0)$ of block $B$ with ground, $B$ may be lifted off ground.$(\text{Take,} k=40 \mathrm{N} {\mathrm{m}}^{-1}\text{,} g=10 \mathrm{m} {\mathrm{s}}^{-2}\text{,} m_{\mathrm{A}}=2 \mathrm{kg}\text{,} m_{\mathrm{B}}=4 \mathrm{kg}\text{.})$

A $3 \mathrm{kg}$ uniform rod rotates in a vertical plane about a smooth pivot at $B$. A spring of constant $k=300 {\mathrm{Nm}}^{-1}$ and of unstretched length $100 \mathrm{mm}$ is attached to the rod as shown. Knowing that in the position shown the rod has an angular velocity of $4 \mathrm{rad} {\mathrm{s}}^{-1}$ clockwise, determine the angular velocity of the rod after it has rotated through. $[g=10 \mathrm{m} {\mathrm{s}}^{-2} ]$

(a) $90º$        (b) $180º$

A carpet of mass $M$ made of inextensible material is rolled along its length in the form of a cylinder of radius $R$and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it. Calculate the horizontal velocity of the axis of the cylindrical part of the carpet when its radius reduces to $R/2$.