(C) Since the force applied to pull the string is radial,the torque acting on the object about the center of the circle is zero. Therefore,the angular

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$\left(\frac{r_1}{r_2}\right)^2$

(C) Since the force applied to pull the string is radial,the torque acting on the object about the center of the circle is zero. Therefore,the angular momentum of the object is conserved.$L = m v_1 r_1 = m v_2 r_2$$\therefore v_2 = v_1 \left(\frac{r_1}{r_2}\right)$Now,the ratio of the new kinetic energy $(K_2)$ to the original kinetic energy $(K_1)$ is:$\frac{K_2}{K_1} = \frac{\frac{1}{2} m v_2^2}{\frac{1}{2} m v_1^2} = \left(\frac{v_2}{v_1}\right)^2$Substituting the value of $v_2/v_1$ from the angular momentum equation:$\frac{K_2}{K_1} = \left(\frac{r_1}{r_2}\right)^2$

Class 11 Physics System of Particles and Rotational Motion Conservation of angular momentum (combined translation and rotational motion)

Medium

$A$ small object is attached to a light string which passes through a hollow tube. The tube is held by one hand and the string by the other. The object is set into rotation in a circle of radius $r_1$. The string is then pulled down,shortening the radius of the circle to $r_2$. The ratio of the new kinetic energy to the original kinetic energy is

A
$\frac{r_1}{r_2}$
B
$1$
C
$\left(\frac{r_1}{r_2}\right)^2$
D
$\left(\frac{r_2}{r_1}\right)^2$

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Class 11

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System of Particles and Rotational Motion

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Conservation of angular momentum (combined translation and rotational motion)

$A$ person is sitting on a rotating stool with their arms outstretched. Suddenly, they fold their arms inward.

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$A$ circular platform of moment of inertia $100 \ kg\ m^2$ with a boy of mass $20 \ kg$ standing at the centre completes one rotation in $10 \ s$. If he walks $2 \ m$ away from the centre along the radius,then one rotation (with respect to the ground) will be completed in $.... \ s$.

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$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.
$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.

EasyAIIMS 2004

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$A$ particle of mass $m$ moves with a velocity $v_0$ in a circle of radius $R_0$ on a smooth horizontal plane. The mass is tied to a string passing through a hole in the smooth plane as shown in the figure. The tension in the string is gradually increased,and finally,the mass $m$ moves in a circle of radius $\frac{R_0}{2}$. What is the ratio of the final kinetic energy to the initial kinetic energy?

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Two discs with moments of inertia $I_1$ and $I_2$ and angular speeds $\omega_1$ and $\omega_2$ are rotating about collinear axes passing through their centers of mass and perpendicular to their planes. If they are joined together to rotate about the same axis,the rotational kinetic energy of the system will be:

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$A$ person is sitting on a rotating stool with their arms outstretched. Suddenly, they fold their arms inward.

$A$ circular platform of moment of inertia $100 \ kg\ m^2$ with a boy of mass $20 \ kg$ standing at the centre completes one rotation in $10 \ s$. If he walks $2 \ m$ away from the centre along the radius,then one rotation (with respect to the ground) will be completed in $.... \ s$.

$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.

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$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.
$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.