HPAS Maths Optional PYQs: Dynamics

On that page, you will find year-wise and question-wise solutions.

Dynamics questions focus on the motion of particles and rigid bodies, including Simple Harmonic Motion (SHM), projectile motion, resisted motion, and central orbits.

HPAS 2024 Dynamics Questions

Question 6(a): Find the work done by the force \(\vec{F}=(x^2-y^2)\mathbf{i}+(x+y)\mathbf{j}\) in moving a particle along the closed path \(C\) containing the curves \(x+y=0\), \(x^2+y^2=16\), and \(y=x\) in the first and the fourth quadrants.

Question 7(a): A particle is projected in a plane with velocity \(\sqrt{\frac{\mu}{3a^6}}\) at a distance \(a\) from the center of force, attracting according to the law \(\frac{\mu}{r^7}\), in a direction inclined at \(30^\circ\) to the radius vector. Show that the orbit is \(r^2 = 2a^2 \cos(2\theta)\).

Question 8(b): The amplitude of a simple harmonic oscillator is doubled. How does this affect the time period, total energy, and maximum velocity of the oscillator?

HPAS 2023 Dynamics Questions

Question 1(e): A particle executes Simple Harmonic Motion with a period of 10 seconds and an amplitude of 5 cm. Calculate the maximum velocity.

Question 8(a): A particle performing Simple Harmonic Motion has a mass of 2.5 gm and a frequency of vibration of 10 Hz. It is oscillating with an amplitude of 2 cm. Calculate the total energy of the particle.

Question 8(b): The motion of a particle under the influence of a central force is described by \(r = a \sin\theta\). Find an expression for the force.

HPAS 2021 Dynamics Questions

Question 7(c): Show that the only law for a central attraction, for which the velocity in a circle at any distance is equal to the velocity acquired in falling from infinity to that distance, is that of the inverse cube.

Question 8(b): A particle of mass \(m\) is attached to a light wire which is stretched tightly between two fixed points with a tension \(T\). If \(a\) and \(b\) are the distances of the particle from the two ends, then show that the period of the small transverse oscillation of \(m\) is \(2\pi\sqrt{\frac{mab}{T(a+b)}}\).

HPAS 2020 Dynamics Questions

Question 1(d): Find the velocity of a particle moving on the surface of a right circular cylinder of radius \(b\).

Question 5(b): A particle of mass \(m\) moves with a central attractive force \(\mu(r^5-c^4r)\) towards the origin. It is projected from an apse at distance \(c\) with velocity \(\sqrt{\frac{2\mu}{3}}c^3\). Show that the equation of the central orbit is \(x^4+y^4=c^4\).

Question 7(a): The tangential acceleration of a particle moving along a circle of radius \(a\) is \(\lambda\) times the normal acceleration. If its speed at a certain time is \(u\), then prove that it will return to the same point after a time \(\frac{a}{\lambda u}(1-e^{-2\pi\lambda})\).

HPAS 2018 Dynamics Questions

Question 8(a): With usual notations, prove that the angular acceleration in the direction of motion of a point moving in a plane is \(\frac{v}{\rho}\frac{dv}{ds} – \frac{v^2}{\rho^2}\frac{d\rho}{ds}\).

HPAS 2017 Dynamics Questions

Question 8(a): A particle moves in a curve such that its tangential and normal accelerations are equal and the angular velocity of the tangent is constant. Find the path.

Question 8(b): A particle describes an ellipse under a force \(\frac{\mu}{(\text{distance})^2}\) towards a focus. If it was projected with velocity \(V\) from a point at a distance \(r\) from the center of force, show that its periodic time is \(\frac{2\pi}{\sqrt{\mu}} \left[ \frac{2}{r} – \frac{V^2}{\mu} \right]^{-3/2}\).

HPAS 2016 Dynamics Questions

Question 8(b): A particle moves in a plane in such a manner that its tangential and normal accelerations are always equal and its velocity varies as \(e^{\tan^{-1}(s/c)}\), \(s\) being the length of the arc of the curve measured from a fixed point on the curve. Find the path.

HPAS 2015 Dynamics Questions

Question 8(b): A particle is moving vertically downwards from rest through a medium whose resistance varying as velocity, discuss its motion.

Question 8(c): The greatest and least velocities of a certain planet in its orbit round the sun are 30 and 29.2 km/sec. Find the eccentricity of the orbit.

HPAS 2014 Dynamics Questions

Question 8(b): A particle moves with simple harmonic motion in a straight line. If in the first second after starting from rest it travels a distance \(a\) and in the next second it travels a distance \(b\) in the same direction, then find the amplitude and period of the motion.

Question 8(c): A particle of mass \(m\) is falling under gravity through a medium whose resistance is \(\mu\) times the velocity. If the particle is released from rest, show that the distance fallen through in time \(t\) is \(\frac{gm^{2}}{\mu^{2}}(e^{\frac{\mu t}{m}}-1-\frac{\mu t}{m})\).