Shortcut Methods and Tricks for Numerical Problems in Physics

Linear Motion in a Plane (1D Motion)

Speed: Speed can be calculated by dividing the distance traveled by the time taken. To quickly calculate speed when distance is given in kilometers and time in hours, multiply the distance by 1000/3600. For example, if a car travels 120 kilometers in 2 hours, its speed is (120 x 1000/3600) = 33.33 m/s.
Acceleration: Acceleration can be determined by dividing the change in velocity by the time interval. To simplify calculations when acceleration is given in m/s² and time in seconds, multiply acceleration by the time squared. For instance, if an object starts from rest and accelerates at 5 m/s² for 10 seconds, its final velocity is 5 x (10²) = 50 m/s.

Angular Motion in a Plane (2D Motion)

Angular Speed: Angular speed can be calculated by dividing the angular displacement by the time taken. To easily convert degrees to radians, multiply the angle by π/180. For example, if a wheel rotates 120 degrees in 2 seconds, its angular speed is (120 x π/180) / 2 = 1.05 rad/s.
Torque: Torque can be calculated by multiplying the force applied by the perpendicular distance from the axis of rotation. Using the right-hand rule, determine the direction of the torque. For instance, if a force of 20 N is applied at a perpendicular distance of 0.5 m from the axis of rotation, the torque is 20 x 0.5 = 10 N·m.

Projectile Motion:

Range: The range (horizontal distance covered by a projectile) for an object launched at an angle θ can be calculated using the formula R = (V₀²) * sin(2θ) / g. Here, V₀ is the initial velocity, g is the acceleration due to gravity (approximately 9.81 m/s²), and θ is the angle of projection. For instance, if a ball is thrown with an initial velocity of 20 m/s at an angle of 30 degrees, its range is [(20²) * sin(60)] / 9.81 = 34.2 m.

Uniform Circular Motion:

Period: For an object moving in a circular path with radius ‘R’ and constant angular velocity ‘ω’, the period (T) of its motion is given by T = 2πR/ω. For instance, if a satellite orbits the Earth at a distance of 6378 km with an angular velocity of 7.27 x 10-5 rad/s, its orbital period is 2π x (6.378 x 10³) / (7.27 x 10-5) = 86,164 seconds or approximately 1 day.

Simple Harmonic Motion:

Frequency: The frequency (f) of a simple harmonic oscillator can be calculated using the formula f = 1/T, where T is the period of oscillation. For example, if a pendulum swings back and forth with a period of 2 seconds, its frequency is 1 / 2 = 0.5 Hz.

Damped Harmonic Motion:

Damping Ratio: The damping ratio (ζ) of a damped harmonic oscillator is a dimensionless quantity representing the level of damping. It can be calculated using the formula ζ = c / (2√(mk)), where c is the damping coefficient, m is the mass, and k is the spring constant. For instance, if a vibrating system has a damping coefficient of 0.05 kg/s, a mass of 2 kg, and a spring constant of 20 N/m, its damping ratio is 0.05 / (2√(2 x 20)) ≈ 0.112.

Forced Harmonic Motion:

Resonance: Resonance occurs when the driving force frequency equals the natural frequency of the system. This can be observed as a significant amplitude increase in the system. To find the resonant frequency, use the formula ω = √(k/m), where k is the spring constant and m is the mass. For instance, consider a mass-spring system with a spring constant of 12 N/m and a mass of 0.5 kg. The resonant frequency of this system is √(12 / 0.5) ≈ 3.464 rad/s.