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A solid sphere of mass m = 80 kg and radius r = 0.2 m is released from height h = 5/4 meter. Sphere is initially rotating about horizontal axis passing through its centre of mass. It hits with a stationary cart of mass M = 200 kg exactly at the centre of cart. The cart can move smoothly on the horizontal surface. The collision between sphere and cart occurs in such a way that sphere reaches at same vertical displacement after collision and falls back onto it again. It is found that sphere starts pure rolling at the end of first collision. The coefficient of friction between sphere and cart is $μ = 0.1$ . Match the statement given in List-I to the values given in List-II. $(g = 10 m / \sec^{2})$
LIST-I LIST-II
P) The minimum length (in meter) of cart to occur second collision with the sphere 1) 172
Q) Initial angular velocity $' ω_{0}'$ (in rad/sec) of sphere on the cart during the process. 2) 2.8
R) Magnitude of work done (in Joule) by sphere on the cart during the process. 3) 156
S) Magnitude of work done (in Joule) by cart on the sphere during the process 4) 19.5
T) 5) 16

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$P \to 2; Q \to 4; R \to 5; S \to 1$

$P \to 1; Q \to 3; R \to 1; S \to 3$

$P \to 3; Q \to 5; R \to 4; S \to 2$

$P \to 4; Q \to 3; R \to 4; S \to 1$

answer is A.

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Detailed Solution

$\Rightarrow v_{0} = \sqrt{2 g h} = 5 m / s$ …………(i)
$\Rightarrow J_{y} = Δ P_{y} = 2 \times 80 \times 5 = 800 N - s$ …………..(ii)
$\Rightarrow J_{x} = Δ P_{x}$
$μ J_{y} = m v_{x}$

$v_{x} = 1 m / s$ …………..(iii)

   $\Rightarrow M v_{C} = J_{x}$ …………..(iv)
   $v_{C} = 0.4 m / s$ …………..(v)
$\Rightarrow t_{0} = \frac{2 \times 5}{10} = 1 \sec$ $\Rightarrow L_{\min} = 2 (1 + 0.4) 1 = 2.8 m$ …………..(vi)
At the time of second collision
$R ω' - 1 = 0.4$
   $ω' = 7 r a d / s$ …………..(vii)
$\Rightarrow \vec{J}' = Δ \vec{L}$
$- J_{x} R = I (ω' - ω_{0})$
$- J_{x} R = I (ω' - ω_{0})$
   $ω_{0} = 19.5 r a d / \sec$ ………………(viii)
$\Rightarrow W_{m \to M} = \frac{1}{2} M v_{C}^{2} = 16 J$ ………………(ix)
   $\Rightarrow W_{M \to m} = \frac{1}{2} I {(ω')}^{2} + \frac{1}{2} m v_{x}^{2} - \frac{1}{2} I ω_{0}^{2} = - 172 J$

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