JEE Main

Inclined plane

Acceleration can be found using: $$a = g\sin\theta$$

where:

• (a) is the acceleration in m/s²
• (g) is the acceleration due to gravity ((\approx 9.8\ m/s²))
• (\theta) is the angle of the incline in degrees.

Uniform circular motion

The formula to calculate centripetal force is: $$F_c = \frac{mv^2}{r}$$ where:

• (F_c )is the centripetal force in Newtons (N)
• (m) is the mass of the object in kilograms (kg)
• (v) is the speed of the object in meters per second (m/s)
• (r) is the radius of the circular path in meters (m).

Simple harmonic motion

The formula for determining the time period of a simple harmonic oscillator is: $$T = 2\pi\sqrt{\frac{m}{k}}$$ where:

• (T) is the time period in seconds (s)
• (m) is the mass of the object in kilograms (kg)
• (k) is the spring constant in newtons per meter (N/m).

Collision of elastic balls

The velocities of two elastic balls before and after collision can be calculated using: $$v_1’ = \frac{(m_1-m_2)v_1 + 2m_2v_2}{m_1+m_2}$$

$$v_2’ = \frac{(m_2-m_1)v_2 + 2m_1v_1}{m_1+m_2}$$

where:

• (v_1) and (v_2) are the initial velocities of the two balls
• (v_1’) and (v_2’) are the final velocities of the two balls
• (m_1) and (m_2) are the masses of the two balls.

Newton’s law of cooling

The rate of cooling of a hot object can be calculated using Newton’s law of cooling: $$\frac{dT}{dt} = -k(T - T_{room})$$ where:

• (T) is the temperature of the object in Celsius (°C)
• (T_{room}) is the room temperature in Celsius (°C)
• (k) is the heat transfer coefficient in watts per square meter kelvin (W/m²K)
• (t) is the time in seconds (s).

JEE Advanced

Inclined plane with friction

The acceleration of a body sliding down an inclined plane with friction can be found by: $$a=g\sin\theta-\mu g\cos\theta$$ where:

• (a) is the acceleration in m/s²
• (g) is the acceleration due to gravity ((\approx 9.8\ m/s²))
• (\theta) is the angle of the incline in degrees.
• (\mu) is coefficient of friction.

Non-uniform circular motion

Calculating centripetal force in non-uniform circular motion: $$F_c = mv^2\left(\frac{dy}{dx} + \frac{dx}{dy}\frac{d^2x}{dt^2}\right)$$ where:

• (F_c )is the centripetal force in Newtons (N)
• (m) is the mass of the object in kilograms (kg)
• (v) is the speed of the object in meters per second (m/s)
• (\frac{dy}{dx},\frac{dx}{dy}, \frac{d^2x}{dt^2}) are derivatives from parametric equation of curve.

Damped harmonic motion

The time period of a damped harmonic oscillator can be determined by: $$T = 2\pi\sqrt{\frac{m}{k}}\sqrt{1-\frac{b^2}{4mk}}$$ where:

• (T) is the time period in seconds (s)
• (m) is the mass of the object in kilograms (kg)
• (k) is the spring constant in newtons per meter (N/m)
• (b) is the damping coefficient in newton-seconds per meter (Ns/m).

Collision of inelastic balls

For inelastic collision, the velocities can be calculated using the formula: $$v_{12}=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$ $$v_1’=\frac{(m_1v_1+m_2v_2)-m_1v_{12}}{m_1}$$

$$v_2’=\frac{(m_1v_1+m_2v_2)-m_2v_{12}}{m_2}$$

where

• (m_1) and (m_2) are the masses of the two objects.
• (v_1) and (v_2) are the initial velocities of the objects.
• (v_{12}) is the velocity of the objects immediately after the collision.
• (v_1’) and (v_2’) are the final velocities of the objects.

Motion in a central force field

The trajectory of a body moving in a central force field can be determined by solving: $$\frac{d^2u}{d\phi^2} - u = -\frac{k}{h^2}u^2$$ where

• (u=1/r)
• (r) is the distance from the origin
• (h) is the angular momentum
• (k) is a constant which depends on force

CBSE Board Exams

Inclined plane, uniform circular motion, simple harmonic motion, collision of elastic balls, and Newton’s law of cooling can be solved using the same formulas as mentioned in JEE Main section.

Tips:

• Practice regularly using these formulas to increase your problem-solving skills.
• Pay attention to the units of measurements and convert them if necessary.
• Make sure to understand the concepts and principles before applying these formulas.