$A$ false balance has unequal arms. An object weighs $X$ when placed in one pan and $Y$ when placed in the other pan. The true weight $W$ of the object is equal to:

Two persons of equal height are carrying a long uniform wooden beam of length $\ell$. They are at a distance of $\frac{\ell}{5}$ and $\frac{\ell}{3}$ from the nearest ends of the rod. The ratio of the normal reaction at their heads is

$A$ rod of mass $M = 1 \ kg$ and length $L = 1 \ m$ is suspended horizontally by two ideal strings as shown in the figure. The strings are attached at a distance of $1/3 \ m$ from each end. First, a mass $m_1$ is suspended from the left end while keeping the rod horizontal. Then, a second mass $m_2$ is suspended from the right end, again keeping the rod horizontal. What is the maximum total mass $(m_1 + m_2)$ that can be suspended in this way while maintaining the horizontal orientation of the rod?

$A$ metal rod of weight $W$ is supported by two parallel knife-edges $A$ and $B$. The rod is in equilibrium in a horizontal position. The distance between the two knife-edges is $r$. The centre of mass of the rod is at a distance $x$ from $A$. The normal reaction on $A$ is

$A$ wheel in uniform motion about an axis passing through its centre and perpendicular to its plane is considered to be in mechanical (translational plus rotational) equilibrium because no net external force or torque is required to sustain its motion. However,the particles that constitute the wheel do experience a centripetal acceleration directed towards the centre. How do you reconcile this fact with the wheel being in equilibrium?
How would you set a half wheel into uniform motion about an axis passing through the centre of mass of the wheel and perpendicular to its plane? Will you require external forces to sustain the motion?

Two men are carrying a uniform bar of length $L$ on their shoulders. The bar is held horizontally such that the younger man gets $(1/4)^{th}$ of the total load. Suppose the younger man is at one end of the bar,what is the distance of the other man from the other end?