Q. No.Two identical bodies \(A\) and \(B\) of equal masses have initial velocities \(\overrightarrow{v_1}=4 \hat{i} ~\text{m/s}\) and \(\overrightarrow{v_2}=4 \hat{j}~ \text{m/s}\) respectively. The body \(A\) has acceleration \(\overrightarrow{a_1}=6 \hat{i}+6 \hat{j}~\text{m/s}^2\) while the acceleration of the other body \(B\) is zero. In what type of path does the centre of mass of two bodies move?
1. circular
2. parabolic
3. straight line
4. elliptical
1. circular
2. parabolic
3. straight line
4. elliptical
Subtopic: Center of Mass |
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The position of centre of mass of three masses \(2~\text{kg}.\) \(3~\text{kg}\) and \(15~\text{kg}\) placed with respected to mid point \((p)\) of normal bisector, as shown in the figure is:
1. \( \left(\dfrac{\sqrt{3}}{4}, 1.25\right)~\)
2. \(\left(\dfrac{\sqrt{3}}{4}, 1.0\right) ~\)
3. \( (0,0) \)
4. \((1.25,0)\)
1. \( \left(\dfrac{\sqrt{3}}{4}, 1.25\right)~\)
2. \(\left(\dfrac{\sqrt{3}}{4}, 1.0\right) ~\)
3. \( (0,0) \)
4. \((1.25,0)\)
Subtopic: Center of Mass |
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Given below are two statements:
In the light of the above statements, choose the correct answer from the options given below:
| Statement I | For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles. |
| Statement II | The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t. to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference. |
| 1. | Both Statement I and Statement II are True |
| 2. | Statement I is True but Statement II is False |
| 3. | Statement I is False but Statement II is True |
| 4. | Both Statement I and Statement II are False |
Subtopic: Center of Mass |
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A body of mass \(14~\text{kg}\) initially at rest explodes and breaks into three fragments of masses in the ratio \(2:2:3\). The two pieces of equal masses fly off perpendicular to each other with a speed of \(18~\text{m/s}\) each. The velocity of the heavier fragment is (in m/s)
1. \(10 \sqrt{2}\)
2. \(12 \sqrt{2}\)
3. \(12\)
4. \(24 \sqrt{2}\)
1. \(10 \sqrt{2}\)
2. \(12 \sqrt{2}\)
3. \(12\)
4. \(24 \sqrt{2}\)
Subtopic: Center of Mass |
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A wheel is rolling on a plane surface. The speed of a particle at the highest point of the rim is \(8 ~\text{m/s.}\) The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be:
1. \(8 ~\text{m/s} \)
2. \(8\sqrt{2} ~\text{m/s} \)
3. \(4 ~\text{m/s} \)
4. \(4\sqrt{2} ~\text{m/s}\)
1. \(8 ~\text{m/s} \)
2. \(8\sqrt{2} ~\text{m/s} \)
3. \(4 ~\text{m/s} \)
4. \(4\sqrt{2} ~\text{m/s}\)
Subtopic: Center of Mass |
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A rod of length \(5L\) is bent right angle keeping one side length as \(2 L.\) The position of the centre of mass of the system: (Consider \(L = 10~\text{cm}\))
1. \(5\hat{i} +8\hat{j}\)
2. \(3\hat{i} +7\hat{j} \)
3. \(2\hat{i} +3\hat{j}\)
4. \(4\hat{i} +9\hat{j} \)
1. \(5\hat{i} +8\hat{j}\)
2. \(3\hat{i} +7\hat{j} \)
3. \(2\hat{i} +3\hat{j}\)
4. \(4\hat{i} +9\hat{j} \)
Subtopic: Center of Mass |
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The center of mass of a thin rectangular plate as shown in figure with sides of length \(a\) and \(b,\) whose mass per unit area \(\sigma\) varies as \(\sigma=\dfrac{\left(\sigma_0 x\right)}{ a b}\) (where \(\sigma_0\) is a constant), would be
1. \(\left(\dfrac{1}{3} a, \dfrac{b}{2}\right) \)
2. \(\left(\dfrac{2}{3} a, \dfrac{b}{2}\right) \)
3. \(\left(\dfrac{a}{2}, \dfrac{b}{2}\right) \)
4. \(\left(\dfrac{2}{3} a, \dfrac{2}{3} b\right) \)
1. \(\left(\dfrac{1}{3} a, \dfrac{b}{2}\right) \)
2. \(\left(\dfrac{2}{3} a, \dfrac{b}{2}\right) \)
3. \(\left(\dfrac{a}{2}, \dfrac{b}{2}\right) \)
4. \(\left(\dfrac{2}{3} a, \dfrac{2}{3} b\right) \)
Subtopic: Center of Mass |
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Consider a circular disc of radius \(20~\text{cm}\) with centre located at the origin. A circular hole of radius \(5~\text{cm}\) is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be
1. \(0.5~\text{cm}\)
2. \(2.0~\text{cm}\)
3. \(1.0~\text{cm}\)
4. \(1.5~\text{cm}\)
1. \(0.5~\text{cm}\)
2. \(2.0~\text{cm}\)
3. \(1.0~\text{cm}\)
4. \(1.5~\text{cm}\)
Subtopic: Center of Mass |
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A uniform thin metal plate of mass \(\mathrm{10~kg}\) is shown in the figure. The plate is formed by removing a \((1~\text{m} \times 1~\text{m})\) square notch from the top of a \((3~\text{m} \times 2~\text{m})\) rectangular sheet. If the ratio of the \(x\)-coordinate to the \(y\)-coordinate of the centre of mass of the remaining plate is \(\dfrac{n}{9},\) the value of \(n\) is:
1. \(45\)
2. \(10\)
3. \(15\)
4. \(20\)
1. \(45\)
2. \(10\)
3. \(15\)
4. \(20\)
Subtopic: Center of Mass |
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In a system two particles of masses \(m_1=3 \) kg and \(m_2=2 \) kg are placed at certain distance from each other. The particle of mass \(m_1\) is moved towards the center of mass of the system through a distance \(2~\text{cm}.\) In order to keep the center of mass of the system at the original position. The particle of mass \(m_2\) should move towards the center of mass by the distance:
1. \(3~\text{cm}\)
2. \(4.5~\text{cm}\)
3. \(5~\text{cm}\)
4. \(7~\text{cm}\)
1. \(3~\text{cm}\)
2. \(4.5~\text{cm}\)
3. \(5~\text{cm}\)
4. \(7~\text{cm}\)
Subtopic: Center of Mass |
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