Mechanics Questions and Answers

A particle is attached to one end of a light inextensible string of length $1.3$ m whose other end is fixed at A. The particle moves in a horizontal circle with the string at $35^\circ$ to the vertical. Find the radius of the circle, in metres, to 2 decimal places.

A particle of mass $2m$ kg on a string of length $L$ moves in a horizontal circle with the string at an angle $\theta$ to the vertical, at angular speed $\omega$. Resolving vertically gives $T\cos\theta = 2mg$ and horizontally gives $T\sin\theta = 2m\omega^2 L\sin\theta$. Which expression for $\omega^2$ follows from these two equations?

A particle of mass $2m$ kg hangs from a light inextensible string of length $1.3$ m attached to a fixed point A. It moves at constant speed in a horizontal circle with the string at $35^\circ$ to the vertical. Taking $g = 9.81\,\mathrm{ms^{-2}}$, calculate its angular speed in $\mathrm{rad\,s^{-1}}$, to 2 significant figures.

A uniform rectangular lamina ABCD has a mass of $10$ kg. Taking $g = 9.81\,\mathrm{ms^{-2}}$, state the weight of the lamina in newtons.

A uniform rectangular lamina ABCD has $AD = BC = 2a$ and $AB = DC = a$. Vertex A rests on a horizontal table and a force $P$ acts along the side DC. When moments are taken about A, what is the perpendicular distance from A to the line of action of $P$?

A uniform rectangular lamina ABCD of mass $10$ kg has $AD = BC = 2a$ and $AB = DC = a$. It rests in a vertical plane with A on a rough horizontal table and AD inclined at $30^\circ$ to the horizontal. A force $P$ along DC maintains equilibrium. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find $P$ in newtons, correct to 2 decimal places.

A uniform rectangular lamina of mass $10$ kg rests with one vertex A on a horizontal table. A force of $30.22$ N acts along a side that makes an angle of $30^\circ$ with the vertical, so its vertical component is upward. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find the normal reaction at A in newtons, to 2 decimal places.

A small particle is threaded on a smooth circular wire of radius $0.5$ m fixed in a vertical plane. It starts from rest at the highest point of the wire. Through what vertical height, in metres, has it fallen when it reaches the lowest point?

A particle of mass $1$ kg is threaded on a smooth circular wire of radius $0.5$ m fixed in a vertical plane with centre O. It is slightly disturbed from rest at the highest point A. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find its speed in $\mathrm{ms^{-1}}$ when angle AOP is $90^\circ$, to 2 decimal places.

A particle of mass $1$ kg is threaded on a smooth circular wire of radius $0.5$ m fixed in a vertical plane. It is slightly disturbed from rest at the highest point. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find its speed in $\mathrm{ms^{-1}}$ at the lowest point, to 2 decimal places.

A particle of mass $1$ kg threaded on a smooth circular wire of radius $0.5$ m reaches the lowest point of the wire with $v^2 = 2g$. Which expression gives the reaction $R$ between the wire and the particle there?

A particle of mass $2$ kg hangs at rest from a spring of natural length $1$ m and modulus of elasticity $\lambda$, fixed at O. At rest the stretch is $0.1$ m. Find $\lambda$ in terms of $g$.

A particle of mass $2$ kg hangs from a spring of natural length $1$ m and modulus $20g$, fixed at O, and rests with a stretch of $0.1$ m. It is then pulled down a further $x$ metres. Which expression gives the tension in the spring in that pulled-down position?

A particle performs simple harmonic motion satisfying $\ddot{x} = -10gx$. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find its angular frequency $\omega$ in $\mathrm{rad\,s^{-1}}$, to 2 decimal places.

A particle performs simple harmonic motion satisfying $\ddot{x} = -10gx$. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find the period of the motion in seconds, to 2 decimal places.

A smooth sphere A of mass $2$ kg moving at $4\,\mathrm{ms^{-1}}$ on a smooth horizontal plane collides directly with a stationary smooth sphere B of mass $3$ kg. The coefficient of restitution is $\dfrac14$. Find the speed of B immediately after the impact, in $\mathrm{ms^{-1}}$.

A smooth sphere A of mass $2$ kg moving at $4\,\mathrm{ms^{-1}}$ collides directly with a stationary smooth sphere B of mass $3$ kg, the coefficient of restitution being $\dfrac14$. After the impact B moves off at $2\,\mathrm{ms^{-1}}$. Find the speed of A immediately after the impact, in $\mathrm{ms^{-1}}$.

A smooth sphere B of mass $3$ kg strikes a fixed vertical wall at right angles, moving at $2\,\mathrm{ms^{-1}}$. The coefficient of restitution is $\dfrac14$. Find the speed at which B rebounds, in $\mathrm{ms^{-1}}$.