Question

A system of two light smooth pulleys and light inextensible strings, has two masses M₁ and M₂ , attached as shown. The pulley P₂ is movable and pulley P₁ is a fixed pulley. If the mass M₂ moves up and the mass M₁ moves down, the accelerations of the masses M₁ and M₂ , are, respectively

(1) \((\frac{4M_1-2M_2}{4M_1+M_2})\)g and \((\frac{2M_1-M_2}{4M_1+M_2})g\)

(2) \((\frac{4M_1-2M_2}{4M_1+M_1})\) g and \((\frac{2M_1-M_2}{4M_1+M_2})g\)

(3)  \((\frac{4M_1-2M_2}{4M_1+M_1})\) and \((\frac{2M_1-M_2}{4M_1+M_2})g\)

(4) \((\frac{4M_1-2M_2}{4M_1+M_2})\) g and   \((\frac{2M_1-M_2}{4M_1+M_2})g\)

Answer 1

 (1) \((\frac{4M_1-2M_2}{4M_1+M_2})\)g and \((\frac{2M_1-M_2}{4M_1+M_2})g\)

Let T₁ and T₂ be the tensions in the strings, as shown. We see that

T₁ = 2T₂

Pulley P₂ is massless. If there is a net force on it, acceleration will approach infinity. For the mass M₂ , moving upward, the acceleration will be half that of the mass M₁ , moving downwards. [This is because a downward displacement Δx , of mass M₁ will result in an upward displacement of (Δx/2) of mass M₂]

Hence the equations of motion of the two masses are

Adding (3) and (4), we get

The acceleration of mass M₁ is, therefore,