List of practice Questions

The center of mass of a thin rectangular plate (fig - x) with sides of length $ a $ and $ b $, whose mass per unit area ($ \sigma $) varies as $ \sigma = \sigma_0 \frac{x}{ab} $ (where $ \sigma_0 $ is a constant), would be

Two iron solid discs of negligible thickness have radii $ R_1 $ and $ R_2 $ and moment of inertia $ I_1 $ and $ I_2 $, respectively. For $ R_2 = 2R_1 $, the ratio of $ I_1 $ and $ I_2 $ would be $ \frac{1}{x} $, where $ x $ is:

The temperature of a body in air falls from $ 40^\circ \text{C} $ to $ 24^\circ \text{C} $ in 4 minutes. The temperature of the air is $ 16^\circ \text{C} $. The temperature of the body in the next 4 minutes will be:

A tube of length 1m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is $ F $ and the angular velocity of the tube is $ \omega $, then the value of $ \alpha $ is ______ in SI units.

The moment of inertia of a uniform rod of mass $ m $ and length $ l $ is $ \alpha $ when rotated about an axis passing through the centre and perpendicular to the length. If the rod is broken into equal halves and arranged as shown, then the moment of inertia about the given axis is:

A disc is performing pure rolling if the speed of the top point is 8 m/s. Find the speed of point B.

A rod of linear mass density $ \lambda $ and length $ L $ is bent to form a ring of radius $ R $. Moment of inertia of the ring about any of its diameter is:

A thin solid disk of 1 kg is rotating along its diameter axis at the speed of 1800 rpm. By applying an external torque of $25\pi$ Nm for 40s, the speed increases to 2100 rpm. The diameter of the disk is ______ m.

A solid sphere with uniform density and radius $ R $ is rotating initially with constant angular velocity ($ \omega_1 $) about its diameter. After some time during the rotation, it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius becomes $ \frac{R}{2} $ is $ \omega_2 $. The value of $ x $ is ________.

A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is

Which of the following are correct expression for torque acting on a body?
A. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{L}}$
B. $\ddot{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\ddot{\mathrm{r}} \times \ddot{\mathrm{p}})$
C. $\ddot{\tau}=\ddot{\mathrm{r}} \times \frac{\mathrm{d} \dot{\mathrm{p}}}{\mathrm{dt}}$
D. $\ddot{\tau}=\mathrm{I} \dot{\alpha}$
E. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{F}}$
( $\ddot{r}=$ position vector; $\dot{\mathrm{p}}=$ linear momentum; $\ddot{\mathrm{L}}=$ angular momentum; $\ddot{\alpha}=$ angular acceleration; $\mathrm{I}=$ moment of inertia; $\ddot{\mathrm{F}}=$ force; $\mathrm{t}=$ time $)$
Choose the correct answer from the options given below:

If $\overrightarrow{\mathrm{L}}$ and $\overrightarrow{\mathrm{P}}$ represent the angular momentum and linear momentum respectively of a particle of mass ' $m$ ' having position vector $\overrightarrow{\mathrm{r}}=\mathrm{a}(\hat{\mathrm{i}} \cos \omega \mathrm{t}+\hat{\mathrm{j}} \sin \omega \mathrm{t})$. The direction of force is

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{\mathrm{x}}{5}}$ where $\mathrm{x}=$ _______.

M and R be the mass and radius of a disc. A small disc of radius R/3 is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the centre O and perpendicular to the plane of disc is $ \frac{4}{x} MR^2 $. The value of x is ____.

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of the wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m, will be:

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Electromagnetic waves carry energy but not momentum.
Reason (R): Mass of a photon is zero.
In the light of the above statements, choose the most appropriate answer from the options given below:

The output of the circuit is low (zero) for:

(A) $ X = 0, Y = 0 $
(B) $ X = 0, Y = 1 $
(C) $ X = 1, Y = 0 $
(D) $ X = 1, Y = 1 $
Choose the correct answer from the options given below:

Arrange the following in the ascending order of wavelength ($ \lambda $):
(A) Microwaves ($ \lambda_1 $)
(B) Ultraviolet rays ($ \lambda_2 $)
(C) Infrared rays ($ \lambda_3 $)
(D) X-rays ($ \lambda_4 $) \text{Choose the most appropriate answer from the options given below:}

Three infinitely long wires with linear charge density $ \lambda $ are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?

A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is:

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of $ P_1 $ and $ P_2 $ are orthogonal to each other. The polarizer $ P_3 $ covers both the slits with its transmission axis at $ 45^\circ $ to those of $ P_1 $ and $ P_2 $. An unpolarized light of wavelength $ \lambda $ and intensity $ I_0 $ is incident on $ P_1 $ and $ P_2 $. The intensity at a point after $ P_3 $, where the path difference between the light waves from $ S_1 $ and $ S_2 $ is $ \frac{\lambda}{3} $, is:

The output of the circuit is low (zero) for:

(A) $ X = 0, Y = 0 $
(B) $ X = 0, Y = 1 $
(C) $ X = 1, Y = 0 $
(D) $ X = 1, Y = 1 $
Choose the correct answer from the options given below:

Arrange the following in the ascending order of wavelength ($ \lambda $):
(A) Microwaves ($ \lambda_1 $)
(B) Ultraviolet rays ($ \lambda_2 $)
(C) Infrared rays ($ \lambda_3 $)
(D) X-rays ($ \lambda_4 $)
Choose the most appropriate answer from the options given below:

A parallel plate capacitor with plate area $ A $ and plate separation $ d $ has a capacitance $ C_0 $. A slab of dielectric constant $ K $ having area $ A $ and thickness $ \left(\frac{d}{4}\right) $ is inserted in the capacitor, parallel to the plates. Find the new value of its capacitance.

Given below are two statements:

Statement (I):
are isomeric compounds.

Statement (II):
are functional group isomers.

In the light of the above statements, choose the correct answer from the options given below: