A particle of mass \(m\) moves in the\(XY\) plane with a velocity of \(v\) along the straight line \(AB.\) If the angular momentum of the particle about the origin \(O\) is \(L_A\) when it is at \(A\) and \(L_B\) when it is at \(B,\) then:
1. | \(L_A>L_B\) |
2. | \(L_A=L_B\) |
3. | The relationship between \(L_A\) and \(L_B\) depends upon the slope of the line \(AB.\) |
4. | \(L_A<L_B\) |
A wheel with a radius of \(20\) cm has forces applied to it as shown in the figure. The torque produced by the forces of \(4\) N at \(A\), \(8~\)N at \(B\), \(6\) N at \(C\), and \(9~\)N at \(D\), at the angles indicated, is:
1. \(5.4\) N-m anticlockwise
2. \(1.80\) N-m clockwise
3. \(2.0\) N-m clockwise
4. \(3.6\) N-m clockwise
If the radius of the earth is suddenly contracted to half of its present value, then the duration of the day will be of:
1. | 6 hours | 2. | 12 hours |
3. | 18 hours | 4. | 24 hours |
1. | \({7 \over 3}~\text{m}\) | 2. | \({10 \over 7}~\text{m}\) |
3. | \({12\over 7}~\text{m}\) | 4. | \({9 \over 7}~\text{m}\) |
The centre of the mass of \(3\) particles, \(10\) kg, \(20\) kg, and \(30\) kg, is at \((0,0,0)\). Where should a particle with a mass of \(40\) kg be placed so that its combined centre of mass is \((3,3,3)\)?
1. \((0,0,0)\)
2. \((7.5, 7.5, 7.5)\)
3. \((1,2,3)\)
4. \((4,4,4)\)
The position of a particle is given by \(\vec r = \hat i+2\hat j-\hat k\) and momentum \(\vec P = (3 \hat i + 4\hat j - 2\hat k)\). The angular momentum is perpendicular to:
1. | X-axis |
2. | Y-axis |
3. | Z-axis |
4. | Line at equal angles to all the three axes |
A rigid body rotates about a fixed axis with a variable angular velocity equal to \(\alpha\) \(-\) \(\beta t\), at the time \(t\), where \(\alpha , \beta\) are constants. The angle through which it rotates before it stops is:
1. | \(\frac{\alpha^{2}}{2 \beta}\) | 2. | \(\frac{\alpha^{2} -\beta^{2}}{2 \alpha}\) |
3. | \(\frac{\alpha^{2} - \beta^{2}}{2 \beta}\) | 4. | \(\frac{\left(\alpha-\beta\right) \alpha}{2}\) |
Four particles of mass \(m_1 = 2m\), \(m_2=4m\), \(m_3 =m \), and \(m_4\) are placed at the four corners of a square. What should be the value of \(m_4\) so that the centre of mass of all the four particles is exactly at the centre of the square?
1. | \(2m\) | 2. | \(8m\) |
3. | \(6m\) | 4. | None of these |
Which of the following will not be affected if the radius of the sphere is increased while keeping mass constant?
1. | Moment of inertia | 2. | Angular momentum |
3. | Angular velocity | 4. | Rotational kinetic energy |
A thin uniform circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with an angular velocity . Another disc of the same dimensions but of mass \(\frac{1}{4}M\) is placed gently on the first disc co-axially. The angular velocity of the system will be:
1. | 2. | ||
3. | 4. |