Mechanics Questions

$ABC$ is a triangle, right-angled at $A$. The resultant of the forces acting along $AB$ and $AC$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is a force along $AD$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. The magnitude of the resultant is:

A mass of $M\text{kg}$ is suspended by a weightless string. The horizontal force $F$ that is required to displace it until the string makes an angle of $45^{\circ }$ with the initial vertical direction is:

A player caught a cricket ball of mass $150\text{g}$ moving at a rate of $20\text{m/s}$. If the catching process is completed in $0.1\text{s}$, the force of the blow exerted by the ball on the hand of the player is equal to:

Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?

The potential energy of a $1\text{kg}$ particle free to move along the $x$ -axis is given by $V(x)=\left(\frac{x^4}{4}-\frac{x^2}{2}\right)\text{J}$. The total mechanical energy of the particle is $2\text{J}$. Then, the maximum speed (in m/s) is:

A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1,-1,0)$ is:

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity ${\omega }^{\prime }$:

Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is: