A balloon and its content having mass M is moving up with an acceleration ‘a’. The mass that must be released from the content so that the balloon starts moving up with an acceleration ‘3a’ will be

(Take ‘g’ as acceleration due to gravity)

A $$1 \mathrm{~kg}$$ mass is suspended from the ceiling by a rope of length $$4 \mathrm{~m}$$. A horizontal force '$$F$$' is applied at the mid point of the rope so that the rope makes an angle of $$45^{\circ}$$ with respect to the vertical axis as shown in figure. The magnitude of $$F$$ is :

(Assume that the system is in equilibrium and $$g=10 \mathrm{~m} / \mathrm{s}^2$$)

A heavy iron bar, of weight $$W$$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $$\theta$$ with the horizontal. The weight experienced by the person is :

A light unstretchable string passing over a smooth light pulley connects two blocks of masses $$m_1$$ and $$m_2$$. If the acceleration of the system is $$\frac{g}{8}$$, then the ratio of the masses $$\frac{m_2}{m_1}$$ is :