Practice Problem in Physics for the JEE Main and Advanced

Abhay Kumar

Chapter 7

Rotational and Rolling Motion - all with Video Answers

Educators

Section 1

Section A

Problem 1

Calculate the moment of incrtia of a uniform straight rod $(m, l)$ about the axis passing through its one end and perpendicular to the rod.

Ravindra Yadav

Ravindra Yadav

Numerade Educator

Problem 2

Calculate the moment of inertia of a uniform straight $\operatorname{rod}(m, l)$ about the axis which is perpendicular biscetor of the rod

Ravindra Yadav

Numerade Educator

Problem 3

Calculate the moment of inertia of the uniform straight rod about the given axis of rotation as in figure.

Ravindra Yadav

Numerade Educator

Problem 4

Calculate the moment of inertia of the uniform straight rod about the given axis of rotation as in figure.

Ravindra Yadav

Numerade Educator

Problem 5

Calculate the moment of inertia of a straight rod having linear mass density $\lambda=\Delta s$ where $\Lambda$ is + ve constant and $s$ is the distance from the left end, about the axis passing through it's c.m. and perpendicular to the rod. (length of rod is $l$ )

Ravindra Yadav

Numerade Educator

Problem 6

Calculate the moment of inertia of a straight rod of length $\lambda$ and having linear mass density $\lambda=\Lambda s$ where $\Lambda$ is $a+$ ve constant and $s$ is the distance from the left end, about the axis passing through its $\mathrm{c} \mathrm{m}$. and makes an angle $\alpha$ with the rod.

Ravindra Yadav

Numerade Educator

Problem 7

Calculate the moment of inertia of a uniform straight rod $(m, l)$ about the axis $A A^{\prime}$ and $B B^{\prime}$ respectively as in figure.

Ravindra Yadav

Numerade Educator

Problem 8

Calculate the moment of incrtia of a uniform circular arc shaped wire about the axis passing through its centre and perpendicular to its planc as in figure.

Ravindra Yadav

Numerade Educator

Problem 9

whether $I_{p}=I_{0}+I_{R}$ or not ?

Ravindra Yadav

Numerade Educator

Problem 10

Calculate the moment of incrtia of a uniform sector of a circular disc about the axis passing through the centre and perpendicular to the plane of the disc.

Ravindra Yadav

Numerade Educator

Problem 11

Calculate the moment of incrtia of a uniform circular disc about its diameter

Ravindra Yadav

Numerade Educator

Problem 12

Calculate the moment of incrtia of a uniform circular disc about its tangent lying in its planc.

Ravindra Yadav

Numerade Educator

Problem 13

A circular hole of radius $R / 2$ is cut from a circular disc of radius $R$. If the remaining mass of the disc is $m$, then find its moment of inertia about the axis $\Lambda A^{\prime}$

Ravindra Yadav

Numerade Educator

Problem 14

A circular hole o[ radius $R / 2$ is cut from a circular disc of radius $R$. If the remaining mass of the disc is $m$, then find its moment of incrtia about the axis $B B^{\prime}$

Ravindra Yadav

Numerade Educator

Problem 15

Calculate the moment of inertia of a uniform rectangular plate of diemensions $(l \times b)$ about the axis passing through its centre of mass and perpendicular to the plane of the plate

Ravindra Yadav

Numerade Educator

Problem 16

Calculate the moment ol incrtia ol a uniform rectangular plate of dicmensions $(l \times b)$ about the axis passing through its onc edge parallcl to the length.

Ravindra Yadav

Numerade Educator

Problem 17

Calculate the moment of incrtia of a unilorm rectangular plate of dicmensions $(l \times b)$ and mass $m$ about the axis passing through its $C \cdot M$ and parallel to its length.

Ravindra Yadav

Numerade Educator

Problem 18

Calculate the moment of inertia of uniform rectangular plate of dimensions $(l \times b)$ about the axis passing through its one corner and perpendicular to it's plane.

Ravindra Yadav

Numerade Educator

Problem 19

The mass o\lceil given \Gamma-shaped uniform wirc framc is $m .$ Calculatc the moment of inerlia of the wire framc about the axis passing through the point $O$ and perpendicular to it's planc

Ravindra Yadav

Numerade Educator

Problem 20

Calculate the moment of inertia of a uniform hollow cylinder of mass $m$, radius $R$ and height $H$ about its own axis.

Ravindra Yadav

Numerade Educator

Problem 21

Calculate the moment of inertia of a uniform hollow cylinder of mass $m$, radius $R$ and height $H$ about the oiven axis as in figure.

Ravindra Yadav

Numerade Educator

Problem 22

Calculate the moment of inertia of a uniform hollow cylinder of mass $m$, radius $R$ and height $H$ about the axis passing through its centre of mass and perpendicular to its height as in figure.

Ravindra Yadav

Numerade Educator

Problem 23

Calculate the moment o? incrtia of a unilorm solid cylinder of mass $m$, basc radius $R$ and height $H$ about its own axis.

Ravindra Yadav

Numerade Educator

Problem 24

Calculate the moment of inertia of a uniform solid cylinder of mass $m_{s}$ base radius $R$ and height $H$ about the given axis as in figure.

Ravindra Yadav

Numerade Educator

Problem 25

Calculate the moment of incrtia of a uniform solid cylinder of mass $m$, basc radius $R$ and height $H$ about the axis passing through it's centre of mass and perpendicular to its hcight as in figure.

Ravindra Yadav

Numerade Educator

Problem 26

Culculate moment of inertia of a uniform solid cone of mass $m$, base radius $R$ and height $H$ about its own axis.

Ravindra Yadav

Numerade Educator

Problem 27

Calculate moment of inertia of a uniform solid cone of mass $m$, base radius $R$ and height $H$ about the given axis as in figure.

Ravindra Yadav

Numerade Educator

Problem 28

Calculate the moment of inertia of a uniform solid sphere of mass of $m$ and radius $R$ about its diameter.

Ravindra Yadav

Numerade Educator

Problem 29

Calculate the moment of inertia of a uniform solid sphere of mass of $m$ and radius $R$ about its tangent.

Ravindra Yadav

Numerade Educator

Problem 30

Calculate the moment of incrtia of a uniform hollow sphere of negligible thickness of mass $m$ and radius $R$ about its diamcter

Ravindra Yadav

Numerade Educator

Problem 31

Calculate the moment of inertia of a uniform cuboid of mass $m$ and of dimension $(l \times b \times h)$ about the axis passing through its $\mathrm{c} \mathrm{m}$ and parallel to its height.

Ravindra Yadav

Numerade Educator

Problem 32

Find the moment of incrtia of a triangular lamina of mass $m$, base $b$ and height $h$ about its base.

Ravindra Yadav

Numerade Educator

Problem 33

Find the moment of inertia of a rectangular lamina of mass $m$ and size $b \times l$ about its diagonal.

Ravindra Yadav

Numerade Educator

Problem 34

Find the moment of incria of a triangular lamina of mass $m$ about the axis of rotation $A B$ as in figure.

Ravindra Yadav

Numerade Educator

Problem 35

Find the moment of inertia of a system in which spherical ball of mass $m$ and radius $r$ attached at the end of a straight rod of mass $M$ and length $l$, if this system is free to rotate about an axis passing through the end of the rod (end of the rod opposite to sphere as in figure)

Ravindra Yadav

Numerade Educator

Problem 36

Calculate moment of inertia of a system of $(2 N+1)$ particles, separated by distance $a$ lying along a straight line about an axis passing through the centre.

Ravindra Yadav

Numerade Educator

Problem 37

A circular lamina of radius $a$ has a surfacc mass density $k x^{2}$, where $x$ is the distance from centre and $k$ is a constant. If the mass of the lamina is $M$, find the moment of inertia of the lamina about an axis through the centre and perpendicular to the lamina.

Ravindra Yadav

Numerade Educator

Problem 38

The uniform disc shown in the figure has a moment of inertia of $0.6 \mathrm{~kg}-\mathrm{m}^{2}$ around the axis that passes through $O$ and is perpendicular to the planc. II a scgment is cut out from the dise as shown, what is the moment of incrtia of the remaining disc ?

Ravindra Yadav

Numerade Educator

Problem 39

Find moment of incrtia of a lamina of mass $M$ in the shapc of an cquilatcral triangle about an axis as shown in figure. The length of cach sidc is $L$.

Ravindra Yadav

Numerade Educator

Problem 40

A rod of uniform cross-scction of mass $M$ and lenghth $L$ is hinged about an end to swing frecly in a vertical planc. However, its density is non uniform and varies lincarly from hinged end to the frec cnd doubling its value. Find the moment of incrtia of the rod, about the rotation axis passing through the hinge point.

Ravindra Yadav

Numerade Educator

Problem 41

Calculate the moment of inertia of hollow sphere (mass $M$ ) of inner radius $R$ and outer radius $2 R$, having constant volume mass density, about its diametric axis.

Ravindra Yadav

Numerade Educator

Problem 42

The angular speed of a whecl is increased from 1200 rpm to $3120 \mathrm{rpm}$. in 16 sccond. (rpm means revolution per minute.) Assuming the acecleration to be uniform,
(a) What is its angular aeceleration ?
(b) How many revolutions docs the whecl make during this time?

Ravindra Yadav

Numerade Educator

Problem 43

A solid body rotates about a fixed axis according to the law $\theta=6 t-2 t^{s}$. Ilere, $\theta$ is in radian and $t$ is seconds. Find :
(a) the mean values of the angular velocity and angular acceleration averaged over the time interval between $t=0$ and the complete stop,
(b) the angular acceleration at the moment when the body stops.

Ravindra Yadav

Numerade Educator

Problem 44

The angular acceleration of a whecl is given by $\alpha=12-t$ wherc $\alpha$ is in $\mathrm{rad} / \mathrm{s}^{2}$ and $t$ in second. If the angular velocity of the whecl is $60 \mathrm{rad} / \mathrm{s}$ at the cnd of 4 sccond, find the angular velocity at the cnd o $[6$ sceond and how many revolutions take placc in these 6 second?

Ravindra Yadav

Numerade Educator

Problem 45

Two step pulleys $P_{1}$ and $P_{2}$ connceted by a cross belt. If the angular acecleration of pullcy $P_{2}$ be $2 \mathrm{rad} / \mathrm{s}^{2}$, find the time required for $A$ to travel $30 \mathrm{~m}$ from rest. Also, find the distance moved by $B$ while $A$ moves $40 \mathrm{~m}$.

Ravindra Yadav

Numerade Educator

Problem 46

A particle of mass $m$ is moving with velocity $\vec{v}$ along a line $y=x+5$. Find the angular momentum of the particle about a perpendicular axis passing through origin $O$.

Ravindra Yadav

Numerade Educator

Problem 47

A particle of mass $m$ is projected from origin $O$ with spced $u$ at an angle $\theta$ with positive $x$ -axis. Positive $y$ -axis is in vertically upward dircction. Find the angular momentum of particle at any time $t$ about $O$ before the particle strikes the ground again.

Ravindra Yadav

Numerade Educator

Problem 48

A particle of mass $m$ is projected from the ground with an initial speed $u$ at an angle $\alpha$. Find the magnitude of its angular momentum at the highest point of its trajectory about the point of projection.

Ravindra Yadav

Numerade Educator

Problem 49

As shown in figure a rod of length $l$ is pivoted about a horizontal and smooth pin through one end. The rod is released from rest in vertical position. Find velocity of the centre of mass of the rod, when rod is inclined anole $A$ from the vertical.

Ravindra Yadav

Numerade Educator

Problem 50

A uniform thin rod of mass $m$ and lenght $l$ is standing on a smooth horizontal surface. $\Lambda$ slight disturbance causes the lower end to slip on the smooth surface and the rod starts falling. Find the velocity of centre of mass of the rod at the instant when it makes an angle $\theta$ with horizontal.

Ravindra Yadav

Numerade Educator

Problem 51

A uniform rod of length $2 a$ is held with onc end resting on a smooth horizontal table making an angle $\alpha$ with the vertical. Show that when the rod is released, its angular velocity when it makes an angle $\theta$ with the vertical is given by
$$
\omega-\left[\frac{6 g}{a} \frac{(\cos \alpha-\cos \theta)}{\left(1+3 \sin ^{2} \theta\right)}\right]^{1 / 2}
$$

Ravindra Yadav

Numerade Educator

Problem 52

A rod of length $L$ and mass $M$ lies on a smooth horizontal surface. $\Lambda$ ball of mass $m$ moving with speed $v$ as shown in figure collides elastically with rod. Find the mass of the ball if it becomes at rest immediately after collision.

Ravindra Yadav

Numerade Educator

Problem 53

A metre stick is held vertically with onc cnd on the floor and is then allowed to [all. Find the specd of the other end when it hits the floor, assuming that the cnd on the floor does not slip.

Ravindra Yadav

Numerade Educator

Problem 54

A uniform rod of length $l$ lies on a smooth horizontal table. A particle moving on the table strikes the rod perpendicularly at an cnd and stops. Find the distanec, travelled by the ecntre of the rod by the time it turns through an angle $\theta$

Ravindra Yadav

Numerade Educator

Problem 55

Two solid bodies rotate about stationary, mutually perpendicular, intersecting axes with constant angular velocities $\omega_{1}$ and $\omega_{2}$. Find the relative angular velocity and relative acceleration of one body with respect to other.

Ravindra Yadav

Numerade Educator

Problem 56

In a simple \Lambdatwood machine, two unequal masses $m_{1}$ and $m_{2}$ are connected by a string going over a clamped rough and heavy pulley of radius $R$ and mass $M$. Find the acceleration of the masses $m_{1}$ and $m_{2}$ and also find the tension in the string.

Ravindra Yadav

Numerade Educator

Problem 57

As in figurc mass $m_{1}$ slides on the smooth horizontal sur[acc. The pullcy is rough and heavy in the form of a cylinder of mass $M$ and radius $R$ and string turns the pulley without slipping. Find the accelcration of cach mass, and tension in cach part o\lceil the string.

Ravindra Yadav

Numerade Educator

Problem 58

$A$ uniform $\operatorname{rod} A B$ is hinged at one end $A$. The rod is kept in the horizontal position by a massless string to point $B$ as shown in figurc. If string is cut, then find the reaction of the hinge on the cnd $A$ of the rod at that instant.

Ravindra Yadav

Numerade Educator

Problem 59

A cylinder of mass $m$ is suspended through two strings wrapped around it as shown in figure. Find
(a) the tension $T$ in the string and
(b) the specd of the cylinder as it falls through a distancc $h$.

Ravindra Yadav

Numerade Educator

Problem 60

A spinning spherical ball having angular velocity $\left(v_{\theta} / 2 r\right)$ is projected on a horizontal rough surface with velocity $v_{i}$. Calculate the velocity of centre of mass when slipping ceases.

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 61

A disc of mass $M$ and radius $R$ can rotate freely about a horizontal shaft $O$ which is located at a distance $r$ from the centre of mass of the disc $C . \Lambda$ ssume that the disc is released from the position shown in figure. Find minimum value of $r$ for which the angular acceleration of the disc is maximum.

Ravindra Yadav

Numerade Educator

Problem 62

A solid sphere is set into motion on a rough horizontal surlace with a lincar velocity $v$ in the forward direction and an angular velocity $v / R$ in the anticlockwisc direction as shown in the Find the lincar specd of the sphere
(a) when it stops rotating and
(b) when slipping finally ceases and pure rolling starts.

Ravindra Yadav

Numerade Educator

Problem 63

A light thread with a body of mass $m$ tiod to its cnd is wound on a uniform solid cylinder of mass $M$ and radius $R$. At the moment $f=0$ the system is set in motion. Calculate the angular velocity of the cylinder at time $t$ and the kinctic energy of the whole system at that time.

Ravindra Yadav

Numerade Educator

Problem 64

In the system shown in figure, the cocficicnt of friction betwocn the body $m_{1}$ and the horizontal planc is cqual to $\mu$, and the pullcy of mass $m$ is assumod to be a uniform disc. At the moment $t=0$ the body $m_{2}$ starts descending. Find the work done by the frictional forecs acting on body $m_{1}$ over the first $t$ scconds alter the beginning of the motion.

Ravindra Yadav

Numerade Educator

Problem 65

Study of rolling body on an inclined rough plane:

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 66

A uniform solid cylinder of radius $R$ rolls over a horizontal plane passing into an inclined plane forming an angle $\alpha$ with the horizontal shown in figure. Find the maximum value of the $v_{0}$ which still permits the cylinder to roll into the inclined plane section without a jump. The rolling is assumed as pure.

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 67

A uniform cy linder of radius $R$ is spinned about its axis to the angular velocity $\omega_{\mathrm{b}}$ and then placed into the contact of a rough vertical wall. The coe/ficicnt of friction betwoen the vertical wall and the cylinder as well as betwecn the horizontal surfacc is cqual to $\mu$. How many turns the cy linder rotated belore it stops?

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 68

A solid ball, initially at rest, is given a sharp impulsc by a cuc. The cuc is held horizontally a distanec $h$ above the central linc as in figure. The cue imparts a specd $v_{0}$ to the ball. It rolls and slides while moving forward and acquires a final speed of $(9 \pi) v_{\circ}$. Show that $h=(4 / 5) R$ where $R$ is the radius o\lceil the ball.

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator