First slide

Application of Newtons laws

Question

Two identical small masses each of mass m are connected by a light inextensible string on a smooth horizontal floor. A constant force F is applied at the mid point of the string as shown in the figure. The acceleration of each mass towards each other is,

Difficult

A

$\frac{F}{2 \sqrt{3} m}$
B

$\frac{\sqrt{3} F}{2 m}$
C

$\frac{2}{\sqrt{3}} \frac{F}{m}$
D

None of these

Solution

Let the tension in the string be T at any angular position $θ$ , the acceleration of each ball along x and y axes be a and a₁ respectively. Writing the equation of motion of m, we obtain
$\begin{array}{l} \sum F_{x} = m a \\ \Rightarrow T \cos θ = m a . . . . . . (i) \\ \sum F_{y} = m a_{1} \end{array}$
$\Rightarrow T \sin θ = m a_{1} . . . . . . . (i i)$
At point P, as it is accelerating with an acceleration a, therefore $F - 2 T \cos θ = m_{p} a$ where m_p= mass of the string at the point $P ≅ 0$
$\begin{array}{l} \Rightarrow F = 2 T sos θ . . . . . . . . (i i i) \\ (2) \div (1) \Rightarrow \tan θ = \frac{a_{1}}{a} \\ \Rightarrow a_{1} = a \tan θ \end{array}$
$where a = \frac{T \cos θ}{m} from (i) .$
$Putting T = \frac{F}{2 \cos θ}, from (iii), we obtain$
$a_{1} = \frac{F}{2 m} \tan θ$
$Putting θ = 30^{\circ} \Rightarrow a_{1} = \frac{F}{2 \sqrt{3} m}$

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