Assignments

c) Two collinear harmonic oscillations x₁ = 8 sin (100πt) and x₂ = 12 sin (96πt) are superposed. Calculate the values of time when the amplitude of the resultant oscillation will be (i) maximum and (ii) minimum.

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Introduction We are given two harmonic oscillations: x₁ = 8 sin(100πt) x₂ = 12 sin(96πt) These are sinusoidal functions with different frequencies. When superposed, they result in a phenomenon known as beats. The amplitude of the resultant wave varies with time due to interference between the two waves. Step 1: Beat Frequency Let the two […]

c) Two collinear harmonic oscillations x₁ = 8 sin (100πt) and x₂ = 12 sin (96πt) are superposed. Calculate the values of time when the amplitude of the resultant oscillation will be (i) maximum and (ii) minimum. Read More »

b) The time period of a simple pendulum, called ‘seconds pendulum’, is 2 s. Calculate the length, angular frequency and frequency of the pendulum. What is the difference between a simple pendulum and a compound pendulum?

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Introduction A seconds pendulum is defined as a simple pendulum whose time period is exactly 2 seconds. In this question, we calculate its length, angular frequency, and frequency. Additionally, we differentiate between a simple and compound pendulum. Given Time period (T) = 2 s Acceleration due to gravity (g) = 9.8 m/s² Step 1: Calculate

b) The time period of a simple pendulum, called ‘seconds pendulum’, is 2 s. Calculate the length, angular frequency and frequency of the pendulum. What is the difference between a simple pendulum and a compound pendulum? Read More »

a) A simple harmonic motion is represented by x(t) = a cos(ωt). Obtain expressions for velocity and acceleration of the oscillator. Also, plot the time variation of displacement, velocity and acceleration of the oscillator.

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Introduction Simple harmonic motion (SHM) is a type of oscillatory motion in which the restoring force is directly proportional to the displacement and acts in the opposite direction. The motion is sinusoidal in time and exhibits characteristics such as constant amplitude and periodicity. Displacement Equation Given the equation of SHM: x(t) = a cos(ωt) where:

a) A simple harmonic motion is represented by x(t) = a cos(ωt). Obtain expressions for velocity and acceleration of the oscillator. Also, plot the time variation of displacement, velocity and acceleration of the oscillator. Read More »

b) A ball of mass 60g is moving due south with a speed of 50 ms⁻¹ at latitude 30ºN. Calculate the magnitude and direction of the coriolis force on the ball.

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Introduction This problem deals with the Coriolis force, an apparent force experienced by moving objects in a rotating frame such as the Earth. The Coriolis effect is especially significant for high-speed or long-distance movements and is calculated using the formula: Fc = 2mωv sin(φ) Where: m = mass of the object (in kg) ω =

b) A ball of mass 60g is moving due south with a speed of 50 ms⁻¹ at latitude 30ºN. Calculate the magnitude and direction of the coriolis force on the ball. Read More »

a) What should be the angular velocity of the earth such that a person of mass 80 kg standing on the earth at the equator would actually fly off the earth? b) A ball of mass 60g is moving due south with a speed of 50 ms⁻¹ at latitude 30ºN. Calculate the magnitude and direction of the coriolis force on the ball.

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Introduction This question deals with two different rotational motion concepts: (a) Rotational dynamics of the Earth to find the critical angular velocity at which objects at the equator would become weightless. (b) Coriolis force, which is an apparent force due to Earth’s rotation, affecting moving bodies in a rotating frame. Part (a): Angular Velocity for

a) What should be the angular velocity of the earth such that a person of mass 80 kg standing on the earth at the equator would actually fly off the earth? b) A ball of mass 60g is moving due south with a speed of 50 ms⁻¹ at latitude 30ºN. Calculate the magnitude and direction of the coriolis force on the ball. Read More »

Titan, a satellite of Saturn, has a mean orbital radius of 1.22 × 10⁹ m. The orbital period of Titan is 15.95 days. Hyperion, another satellite of Saturn, orbits at a mean radius of 1.48 × 10⁹ m. Estimate the orbital period of Hyperion.

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Introduction This problem uses Kepler’s Third Law, which relates the orbital period of a satellite to the radius of its orbit. Specifically, for satellites orbiting the same planet (Saturn in this case), the square of the orbital period is proportional to the cube of the orbital radius: (T₁/T₂)² = (R₁/R₂)³ Given Data Orbital radius of

Titan, a satellite of Saturn, has a mean orbital radius of 1.22 × 10⁹ m. The orbital period of Titan is 15.95 days. Hyperion, another satellite of Saturn, orbits at a mean radius of 1.48 × 10⁹ m. Estimate the orbital period of Hyperion. Read More »

At a crossing a truck travelling towards the north collides with a car travelling towards the east. After the collision the car and the truck stick together and move off at an angle of 30º east of north. If the speed of the car before the collision was 20 ms⁻¹, and the mass of the truck is twice the mass of the car, calculate the speed of the truck before and after the collision.

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Introduction This is a classic two-dimensional collision problem involving conservation of momentum. Since the truck and car stick together after collision, it’s an example of a perfectly inelastic collision. We will use vector components to solve for the unknown initial speed of the truck and the final velocity after the collision. Given Data Mass of

At a crossing a truck travelling towards the north collides with a car travelling towards the east. After the collision the car and the truck stick together and move off at an angle of 30º east of north. If the speed of the car before the collision was 20 ms⁻¹, and the mass of the truck is twice the mass of the car, calculate the speed of the truck before and after the collision. Read More »

A child of mass 50 kg is standing on the edge of a merry go round of mass 250 kg and radius 3.0 m which is rotating with an angular velocity of 3.0 rad s⁻¹. The child then starts walking towards the centre of the merry go round. What will be the final angular velocity of the merry go round when the child reaches the centre?

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Introduction This problem is based on the principle of conservation of angular momentum. When no external torque acts on a system, the total angular momentum remains constant. As the child moves toward the center, the system’s moment of inertia changes, which affects the angular velocity. Given Data Mass of child (m) = 50 kg Mass

A child of mass 50 kg is standing on the edge of a merry go round of mass 250 kg and radius 3.0 m which is rotating with an angular velocity of 3.0 rad s⁻¹. The child then starts walking towards the centre of the merry go round. What will be the final angular velocity of the merry go round when the child reaches the centre? Read More »

A girl is sitting with her dog at the left end of a boat of length 10.0 m. The mass of the girl, her dog and the boat are 60.0 kg, 30.0 kg and 100.0 kg respectively. The boat is at rest in the middle of the lake. Calculate the centre of mass of the system. If the dog moves to the other end of the boat, the girl staying at the same place, how far and in what direction does the boat move?

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Introduction This question is based on the principle of conservation of momentum and center of mass in a system with no external horizontal forces. Since the boat is on a lake, the system is isolated, and the center of mass of the system must remain unchanged in the absence of external forces. Given Data Mass

A girl is sitting with her dog at the left end of a boat of length 10.0 m. The mass of the girl, her dog and the boat are 60.0 kg, 30.0 kg and 100.0 kg respectively. The boat is at rest in the middle of the lake. Calculate the centre of mass of the system. If the dog moves to the other end of the boat, the girl staying at the same place, how far and in what direction does the boat move? Read More »

A horizontal rod with a mass of 10 kg and length 12 m is hinged to a wall at one end and supported by a cable which makes an angle of 30º with the rod at its other end. Calculate the tension in the cable and the force exerted by the hinge.

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Introduction This is a problem involving static equilibrium. The horizontal rod is supported at one end by a hinge and at the other end by a cable inclined at an angle. To solve this, we apply the conditions of equilibrium: Sum of all horizontal forces = 0 Sum of all vertical forces = 0 Sum

A horizontal rod with a mass of 10 kg and length 12 m is hinged to a wall at one end and supported by a cable which makes an angle of 30º with the rod at its other end. Calculate the tension in the cable and the force exerted by the hinge. Read More »

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