Q. No.Two cars \(A\) and \(B\) each of mass \(10^3~\text{kg}\) are moving on parallel tracks separated by a distance of \(10~\text{m}\), in same direction with speeds \(72~\text{km/h}\) and \(36~\text{km/h}\). The magnitude of angular momentum of car \(A\) with respect to car \(B\) is: (in J.s)
1. \(3.6 \times 10^5\)
2. \( 10^5\)
3. \(3 \times 10^5\)
4. \(2 \times 10^5\)
1. \(3.6 \times 10^5\)
2. \( 10^5\)
3. \(3 \times 10^5\)
4. \(2 \times 10^5\)
Subtopic: Angular Momentum |
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When the position vector \(\vec r = x \vec i + y \vec j + z\vec k\) changes sign as \(- \vec r\), which one of the following vector will not flip under sign change?
1. Linear momentum
2. Velocity
3. Acceleration
4. Angular momentum
1. Linear momentum
2. Velocity
3. Acceleration
4. Angular momentum
Subtopic: Angular Momentum |
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A solid sphere with uniform density and radius \(R\) is rotating initially with constant angular velocity \(\omega_1\) about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become \(\dfrac{R}{2} \) is \(x\omega_1.\) The value of \(x \) is:
1. \(20 \)
2. \(32 \)
3. \(40 \)
4. \(16 \)
1. \(20 \)
2. \(32 \)
3. \(40 \)
4. \(16 \)
Subtopic: Angular Momentum |
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If \(\vec{L}\) and \(\vec{p}\) represent the angular momentum and linear momentum respectively of a particle of mass \(m\) having position vector as \(\vec{r}=a( \cos \omega t \hat{i}+ \sin \omega t \hat{j}) \). The direction of force is:
| 1. | Opposite to the direction of \(\vec{L}\times\vec{p}\) |
| 2. | Opposite to the direction of \(\vec{L}\) |
| 3. | Opposite to the direction of \(\vec{r}\) |
| 4. | Opposite to the direction of \(\vec{p}\) |
Subtopic: Angular Momentum |
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An object of mass \(m\) is projected from origin in a vertical \(xy\) plane at an angle \(45^\circ\) with the \(x-\)axis with an initial velocity \(v_0\). The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be
[\(g\) is acceleration due to gravity]
1. \(\dfrac{m v_0^3}{2 \sqrt{2} g}\) along positive \(z-\)axis
2. \(\dfrac{m v_0^3}{4 \sqrt{2} g} \) along positive \(z-\)axis
3. \(\dfrac{m v_0^3}{2 \sqrt{2} g}\) along negative \(z-\)axis
4. \(\dfrac{m v_0^3}{4 \sqrt{2} g}\) along negative \(z-\)axis
[\(g\) is acceleration due to gravity]
1. \(\dfrac{m v_0^3}{2 \sqrt{2} g}\) along positive \(z-\)axis
2. \(\dfrac{m v_0^3}{4 \sqrt{2} g} \) along positive \(z-\)axis
3. \(\dfrac{m v_0^3}{2 \sqrt{2} g}\) along negative \(z-\)axis
4. \(\dfrac{m v_0^3}{4 \sqrt{2} g}\) along negative \(z-\)axis
Subtopic: Angular Momentum |
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Three equal masses \(m\) are kept at vertices \((A, B, C)\) of an equilateral triangle of side \(a\) in free space. At \(t=0,\) they are given an initial velocity
1. \(3 ~{am}V_0\)
2. \(\dfrac{3}{2} {amV}_0\)
3. \(\dfrac{\sqrt{3}}{2} {amV}_0\)
4. \(\dfrac{1}{2} {amV}_0\)
1. \(3 ~{am}V_0\)
2. \(\dfrac{3}{2} {amV}_0\)
3. \(\dfrac{\sqrt{3}}{2} {amV}_0\)
4. \(\dfrac{1}{2} {amV}_0\)
Subtopic: Angular Momentum |
Level 4: Below 35%
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A tube of length \(1~\text m\) is filled completely with an ideal liquid of mass \(2M,\) and closed at both ends. The tube is rotated uniformly in the horizontal plane about one of its ends. If the force exerted by the liquid at the other end is \(F\) then the angular velocity of the tube is \(\sqrt{\dfrac{F}{\alpha M}}\) in SI units. The value of \(\alpha\) is:
1. \(\alpha = 1\)
2. \(\alpha = 2\)
3. \(\alpha = \dfrac{1}{2}\)
4. \(\alpha = 4\)
1. \(\alpha = 1\)
2. \(\alpha = 2\)
3. \(\alpha = \dfrac{1}{2}\)
4. \(\alpha = 4\)
Subtopic: Angular Momentum |
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The position vectors of two \(1~\text{kg}\) particles, \(A\) and \(B,\) are given by \(\vec{r}_A=\left(\alpha_1 t^2 \hat{\imath}+\alpha_2 t \hat{\jmath}+\alpha_3 t \hat{k}\right) m \) and \(\vec{B}_A=\left(\beta_1 t^2 \hat{\imath}+\beta_2 t \hat{\jmath}+\beta_3 t \hat{k}\right) m \) respectively. \(\left(\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{~nm} / \mathrm{s}, \alpha_3=2 \mathrm{~m} / \mathrm{s}, \beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s}\right), \) where \(t\) is time \(n\) and \(p\) are constants. At \( t = 1 s\), \(|\vec V_A|=|\vec V_B|\) are velocities \(\vec V_ A\) and \(\vec V_ B\) of the particles are orthogonal to each other, At \(t=1~\text s,\) the magnitude of angular momentum of particle \(A\) with respect to the position of particle \(B\) is \(\sqrt L ~kgm^2 s^{-1}\). The value of \(L\) is:
1. \(90\)
2. \(100 \)
3. \(64\)
4. \(121\)
1. \(90\)
2. \(100 \)
3. \(64\)
4. \(121\)
Subtopic: Angular Momentum |
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If the radius of the Earth is reduced to three-fourth of its present value without a change in its mass, then the value of the duration of the day of the Earth will be:
1. 22 hours 30 minutes
2. 20 hours 30 minutes
3. 18 hours 30 minutes
4. 13 hours 30 minutes
1. 22 hours 30 minutes
2. 20 hours 30 minutes
3. 18 hours 30 minutes
4. 13 hours 30 minutes
Subtopic: Angular Momentum |
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A particle of mass \(m\) is projected with speed \(v\) at an angle of \(30^\circ\) with the horizontal. When the particle is at the maximum height, its angular momentum about the point of projection is:
| 1. | \(\dfrac{mv^3}{16g}\) | 2. | \(\dfrac{\sqrt3 mv^3}{16g}\) |
| 3. | \(\dfrac{mv^3}{3g}\) | 4. | \(\dfrac{\sqrt3mv^3}{8g}\) |
Subtopic: Angular Momentum |
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