A-Level Further Maths: Further Mechanics Practice

A particle oscillates with SHM of amplitude 0.05 m and angular frequency 10 rad/s. What is its maximum speed?

A mass of 0.4 kg is attached to a spring with spring constant k = 100 N/m. Find the period of oscillation.

In a damped SHM system, the amplitude decays to half its initial value in 5 seconds. If the damping constant b = 0.2 kg/s and mass m = 0.5 kg, what is the initial amplitude if the amplitude after 10 seconds is 0.03 m?

A forced oscillator has natural frequency ω₀ = 20 rad/s and damping constant b = 4 kg/s. A driving force F₀ cos(ωt) is applied. At what driving frequency ω does the amplitude of steady-state oscillation become maximum?

A car of mass 1200 kg travels around a roundabout of radius 25 m at 12 m/s. What is the centripetal force required?

A conical pendulum has a string of length 0.5 m making an angle of 30° with the vertical. Find the angular velocity of the bob.

A bead of mass m slides on a frictionless vertical circular wire of radius r. At the top of the circle, the normal reaction is zero. What is the speed of the bead at this point?

A satellite orbits Earth at altitude 300 km (Earth radius 6370 km, g = 9.81 m/s² at surface). Find its orbital speed.

The gravitational field strength on the surface of a planet of mass M and radius R is g. What is the gravitational field strength at a height R above the surface?

Two identical masses m are placed at points A and B, separated by distance 2d. At the midpoint P, what is the magnitude of the gravitational field strength?

A planet has twice the mass and twice the radius of Earth. What is the escape velocity from its surface compared to Earth’s escape velocity v_E?

Two stars of masses M and 4M orbit their common centre of mass in circular orbits. If their separation is d, find the orbital period of each star.

What are the dimensions of the gravitational constant G in terms of base quantities M, L, T?

A student proposes that the period T of a pendulum is given by T = 2π√(L/g). Check dimensional consistency.

The drag force F on a sphere moving through a fluid depends on the radius r, velocity v, fluid density ρ and viscosity η. If F = kr^a v^b ρ^c η^d, and [η] = ML⁻¹T⁻¹, find the relationship between a, b, c, d.

Using the Buckingham Pi theorem with n = 5 variables (F, r, v, ρ, η) and k = 3 base dimensions, how many dimensionless Pi groups are expected?