LU B.Sc Sem 3 Mathematics Mechanics Important Questions 2026 | PYQ Pattern

Q1. (a) Obtain the Cartesian equation of the common catenary.
(b) A uniform chain of length l hangs from two points A and B on the same level, the tension at each end being n times that at the lowest point. Prove that the span AB is [l/√(n² – 1)] log[n + √(n² – 1)]. [15 Marks]

Q2. (a) Show that any system of forces can be reduced to two forces, the line of action of one of which may be chosen arbitrarily.
(b) What is Poinsot’s central axis? Obtain its vector equation. [15 Marks]

Q3. (a) A solid sphere rests inside a fixed rough hemispherical bowl whose radius is twice that of the sphere. Prove that the equilibrium is stable however heavy a weight is fixed to the top of the sphere.
(b) A cone and a hemisphere on a common base rest with the hemisphere on a rough horizontal table. Prove that for stable equilibrium the height of the cone can be at most √3 times the radius. [20 Marks]

Q4. (a) Five equal weightless rods are jointed to form a rhombus ABCD with diagonal BD. A weight W hangs from C and the system is suspended from A. Prove that the thrust in BD is W/√3.
(b) Two equal uniform rods AB and AC, each 2b long, hinged at A, rest on a smooth vertical circle of radius a. If the angle between them is 2θ, prove that b sin³θ = a cos θ. [20 Marks]

Q5. (a) Obtain the radial and transverse components of velocity and acceleration.
(b) The radial and transverse velocities of a point are λr² and μθ². Find its path and accelerations. [20 Marks]

Q6. (a) A particle in SHM is at distances x₁, x₂, x₃ from the centre at three successive seconds. Prove that its period is 2π / cos⁻¹[(x₁ + x₃)/(2x₂)].
(b) Prove that a particle in SHM takes one-sixth of its period to go from maximum displacement to half the amplitude. [20 Marks]

Q7. A particle slides down a smooth cycloid with vertical axis and vertex downwards. Find its period and show that it takes equal times to fall through the first and second halves of the vertical height. [20 Marks]

Q8. A particle is thrown vertically upwards with speed u in a medium whose resistance varies as the square of the velocity. Find the greatest height reached and show that it returns with speed u₀ given by 1/u₀² = 1/u² + k/g. [20 Marks]

Q9. (a) Obtain the differential equation of a central orbit in polar and pedal forms.
(b) For the central orbit rⁿ = aⁿ cos nθ, find the law of force. [20 Marks]

Q10. (a) Show that the tetrahedron formed by conjugate forces has constant volume.
(b) Obtain the equation of motion of a particle of variable mass. [20 Marks]