Shortcut Methods

I. Kinematics

1. Uniform Circular Motion: To find the tangential velocity (v) of a point on the rim of a rotating disk, simply multiply the angular velocity (\omega) by the radius (r) of the disk.

$$v = \omega r$$

2. Uniform Angular Acceleration: To find the angular displacement (\theta) of a rotating wheel that starts from rest and accelerates uniformly, use the following formula:

$$ \theta = \frac{1}{2} \alpha t^2 $$

January 1, 0001 read

Shortcut Methods

**1.$$ An alpha particle (helium nucleus) of energy 5.5 MeV is scattered through 180° by a gold nucleus. The distance of the closest approach is: Shortcut: The distance of closest approach is given by: $$b = \frac{2Z_1Z_2e^2}{K}$$

Where:

(Z_1) and (Z_2) are the atomic numbers of the two nuclei.
(e) is the elementary charge.
(K) is the kinetic energy of the alpha particle.

Substituting the given values, we get:

$$b = \frac{2(2)(79)(1.6 \times 10^{−19} \text{ C})^2}{5.5 \times 10^6 \text{ eV}}$$ $$b = 30 \times 10^{−15} \text{ m}$$ $$b = 30 \text{ fm}$$

January 1, 0001 read

Shortcut Methods

1. Sum of n terms of an A.P.

$$S_n = \frac{n}{2}[2a_1 + (n - 1)d]$$

where (a_1) is the first term, (d) is the common difference, and (n) is the number of terms.

2. Sum of n terms of a G.P.

$$S_n = \frac{a_1(r^n - 1)}{r - 1}$$

where (a_1) is the first term, (r) is the common ratio, and (n) is the number of terms.

January 1, 0001 read

Shortcut Methods

Number of elements in the power set of a set with n elements: 2^n
- Shortcut: The number of elements in the power set of a set with n elements is equal to the total number of subsets of that set, which is 2^n.
- Example: If a set A = {1, 2, 3}, then the power set of A is P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. Thus, the number of elements in the power set of A is 2^3 = 8.
Number of subsets of a set with n elements: 2^n - 1

January 1, 0001 read

Shortcut Methods

1. Multiplication:

Fast Doubling: To quickly multiply a number by 2, simply shift all its digits one place to the left and add a zero at the end.
Tricks for Numbers Ending with 5:
To multiply a number ending in 5 by any single-digit number, first, multiply the single-digit number by 10, and then add half of that result to itself.
Multiplying by 9, 11, or 12:
Multiplication by 9: To multiply a number by 9, subtract the number from its reverse, and then append a zero to the end.
Multiplication by 11: To multiply a number by 11, add the digits in the odd positions and then add the digits in the even positions. Finally, append these two sums together.
Multiplication by 12: To multiply a number by 12, first multiply it by 11 and then add the number back to itself.

2. Division:

January 1, 0001 read

Shortcut Methods

#(B) Fusion Shortcut Methods and Tricks:

To calculate the energy released by the fusion of two deuterium atoms, use the following formula:

$$E = 3.2\text{ MeV} = 3.2 \times 10^6 \text{ eV} = 5.12 \times 10^{-13} \text{ J}$$

To calculate the energy released by the fusion of two tritium atoms, use the following formula:

$$E = 17.6\text{ MeV} = 17.6 \times 10^6 \text{ eV} = 2.82 \times 10^{-12} \text{ J}$$

January 1, 0001 read

Shortcut Methods

Determine the number of neutrons in an atom of carbon-14 given that its atomic number is 6 and mass number is 14.

Solution:
- The number of neutrons = mass number - atomic number
- = 14 - 6
- = 8 neutrons

Calculate the binding energy per nucleon of oxygen-16, given its mass defect is 0.138 atomic mass units (amu).

Solution:
- Binding energy per nucleon = (Mass defect/Mass number)
- = (0.138 amu / 16 amu)
- = 0.008625 amu/nucleon

A sample of a radioactive substance has a half-life of 10 days. If the initial activity of the sample is 100 decays per minute, what will be its activity after 20 days?

Solution:
- After 1 half-life (10 days), the activity will be 100/2 = 50 decays per minute.
- After another 10 days (20 days total), the activity will be 50/2 = 25 decays per minute.

CBSE Board Exam - Numerical Problems.

An atom has 17 protons and 18 neutrons. What is its atomic number and mass number?

Solution:
- Atomic number = number of protons = 17
- Mass number = number of protons + number of neutrons = 17 + 18 = 35

Calculate the relative atomic mass of an element if it has two isotopes with masses 20 amu and 22 amu, and the respective abundances are 90% and 10%.

Solution:
- Relative atomic mass = (mass of isotope 1 x abundance of isotope 1) + (mass of isotope 2 x abundance of isotope 2)
- = (20 amu x 0.90) + (22 amu x 0.10)
- = 20.2 amu

A radioactive element has a half-life of 5 hours. If 1000 atoms are present initially, how many atoms will remain after 10 hours?

Solution:
- After 1 half-life (5 hours), 1000/2 = 500 atoms will remain.
- After another 5 hours (10 hours total), 500/2 = 250 atoms will remain.

January 1, 0001 read

Shortcut Methods

Shortcuts and Tricks for Electrostatic Problems:

Use symmetry to simplify the problem. Often, the charge distribution or geometry of a problem has some symmetry that can be exploited to simplify the calculation. For example, if the charge distribution is spherically symmetric, then the electric field will be the same in all directions.
Use Gauss’s law to calculate the electric flux through a closed surface. Gauss’s law can be used to calculate the net electric flux through a closed surface without having to know the details of the charge distribution inside the surface. This can be a powerful tool for solving problems involving complex charge distributions.
Use the method of images to solve problems involving conductors and dielectrics. The method of images involves placing imaginary charges in the problem in such a way that the boundary conditions are satisfied. This can be a powerful tool for solving problems involving the interaction of electric fields with conductors and dielectrics.
Use superposition to combine the effects of multiple charges. The principle of superposition states that the net electric field due to a collection of charges is the vector sum of the electric fields due to each individual charge. This can be a useful tool for solving problems involving multiple charges.
Use energy conservation to solve problems involving the motion of charged particles. Energy conservation can be used to determine the velocity and trajectory of charged particles in electric fields. This can be a useful tool for solving problems involving the motion of charged particles in accelerators, ion thrusters, and other devices.

CBSE Board Level

Shortcuts and Tricks for Electrostatic Problems:

Remember the formulas for the electric field due to a point charge and a dipole. These formulas are: $$E = \frac{kq}{r^2}$$ for a point charge and $$E = \frac{1}{4\pi\epsilon_0}\frac{2qs}{r^3}$$ for a dipole.
Use Gauss’s law to calculate the electric flux through a closed surface. Gauss’s law states that the net electric flux through a closed surface is equal to the total charge enclosed by the surface divided by the permittivity of free space.
Use the method of images to solve problems involving conductors and dielectrics. The method of images involves placing imaginary charges in the problem in such a way that the boundary conditions are satisfied.
Use superposition to combine the effects of multiple charges. The principle of superposition states that the net electric field due to a collection of charges is the vector sum of the electric fields due to each individual charge.
Use energy conservation to solve problems involving the motion of charged particles. Energy conservation can be used to determine the velocity and trajectory of charged particles in electric fields.

January 1, 0001 read

Shortcut Methods

1. A glass prism of refractive index 1.52 is immersed in water of refractive index 1.33. A light ray is incident on the prism at an angle of 45 degrees. Calculate the angle of emergence from the prism.

Shortcut Method: Use Snell’s law at each surface of the prism. At the first surface: $$\sin i_1 = n_2 \sin r_1 \Rightarrow \sin 45^\circ = 1.33 \sin r_1 \Rightarrow r_1 = 32.9^\circ$$ At the second surface: $$\sin r_2 = n_1 \sin i_2 \Rightarrow \sin r_2 = 1.52 \sin 32.9^\circ \Rightarrow r_2 = 41.1^\circ$$
Trick: The sum of the incident angle and the emergent angle is equal to the angle of deviation, which is given by: $$\delta = i_1 + e - A \Rightarrow \delta = 45^\circ + 41.1^\circ - 60^\circ = 26.1^\circ$$ Therefore, the angle of emergence is $$e = 41.1^\circ$$

2. A light ray passes from air into a glass block of refractive index 1.52 at an angle of incidence of 42 degrees. Calculate the angle of refraction.

January 1, 0001 read

Shortcut Methods

1. A glass prism of refractive index 1.52 is immersed in water of refractive index 1.33. A light ray is incident on the prism at an angle of 45 degrees. Calculate the angle of emergence from the prism.

Shortcut Method: Use Snell’s law at each surface of the prism. At the first surface: $$\sin i_1 = n_2 \sin r_1 \Rightarrow \sin 45^\circ = 1.33 \sin r_1 \Rightarrow r_1 = 32.9^\circ$$ At the second surface: $$\sin r_2 = n_1 \sin i_2 \Rightarrow \sin r_2 = 1.52 \sin 32.9^\circ \Rightarrow r_2 = 41.1^\circ$$
Trick: The sum of the incident angle and the emergent angle is equal to the angle of deviation, which is given by: $$\delta = i_1 + e - A \Rightarrow \delta = 45^\circ + 41.1^\circ - 60^\circ = 26.1^\circ$$ Therefore, the angle of emergence is $$e = 41.1^\circ$$

2. A light ray passes from air into a glass block of refractive index 1.52 at an angle of incidence of 42 degrees. Calculate the angle of refraction.

January 1, 0001 read

Shortcut Methods

1. Conversion between different units:

For length: Multiply the given value by 1000 to convert kilometers to meters, and divide the given value by 1000 to convert meters to kilometers.
For mass: Multiply the given value by 1000 to convert grams to kilograms, and divide the given value by 1000 to convert kilograms to grams.
For volume: Multiply the given value by 1000 to convert liters to cubic centimeters, and divide the given value by 1000 to convert cubic centimeters to liters.

2. Dimensional analysis:

January 1, 0001 read

Shortcut Methods

1. Finding the focal length of a lens using the mirror equation: Use the mirror equation 1/f = 1/p + 1/q, where f is the focal length, p is the object distance, and q is the image distance, to find the focal length of a lens.

2. Determining magnification in a simple microscope: Use the formula for magnifying power of a microscope, M = (1 + D/f), where M is the magnification, D is the distance of the object from the lens, and f is the focal length of the lens.

January 1, 0001 read

Shortcut Methods

I. Kinematics

Shortcut Methods

1. Sum of n terms of an A.P.

2. Sum of n terms of a G.P.

Shortcut Methods

CBSE Board Exam - Numerical Problems.

Shortcut Methods

Shortcuts and Tricks for Electrostatic Problems:

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Shortcuts and Tricks for Electrostatic Problems:

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Shortcut Methods

I. Kinematics#

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1. Sum of n terms of an A.P.#

2. Sum of n terms of a G.P.#

Shortcut Methods

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CBSE Board Exam - Numerical Problems.#

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Shortcuts and Tricks for Electrostatic Problems:#

CBSE Board Level#

Shortcuts and Tricks for Electrostatic Problems:#

Shortcut Methods

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Medical Preparation

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Important Resources

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NCERT Exercise Video Solution

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For International Students

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I. Kinematics

1. Sum of n terms of an A.P.

2. Sum of n terms of a G.P.

CBSE Board Exam - Numerical Problems.

Shortcuts and Tricks for Electrostatic Problems:

CBSE Board Level

Shortcuts and Tricks for Electrostatic Problems: