Two masses are connected to the ends of massless rope and allowed to move as shown in the figure. The acceleration of the centre of mass assuming pulley is massless and frictionless, is
1.
2. 0
3.
4.
Two persons of mass 55 kg and 65 kg respectively, are at the opposite ends of a boat.The length of the boat is 3.0m and weighs 100 kg.The 55 kg man walks up to the 65 kg man and sits with him.If the boat is in still water the centre of mass of the system shifts by
(1)3.0m
(2)2.3m
(3)zero
(4)0.75m
A man of 50 kg mass is standing in a gravity free space at a height of 10m above the floor. He throws a stone of 0.5 kg mass downwards with a speed When the stone reaches the floor, the distance of the man above the floor will be
1. 9.9m
2. 10.1m
3. 10m
4. 20m
1. | zero | 2. | \(1\) m |
3. | \(2\) m | 4. | \(5\) m |
A man inside a freely falling box throws a heavy ball towards a side wall. The ball keeps on bouncing between the opposite walls of the box. We neglect air resistance and friction. Which of the following figures depicts the motion of the centre of mass of the entire system (man, the ball and the box)?
1. | 2. | ||
3. | 4. |
A circle of radius \(a\) is cut out from a square of uniform density and side \(4a\). If initial mass of the square is \(M\), then the centre of mass of the remaining part is: (where \(O\) is center of square)
1. \(\frac{a}{16-\pi}\) to the left of \(O\).
2. \(\frac{a}{16-\pi}\) to the right of \(O\).
3. \(\frac{\pi a}{16-\pi}\) to the right of \(O\).
4. \(\frac{\pi a}{16-\pi}\) to the left of \(O\).
Three masses \(m,\) \(2m,\) and \(3m\) are thrown from the top of a tower such that \(m\) is thrown vertically upward with \(10\) m/s, \(2m\) is thrown horizontally with \(15\) m/s and \(3m\) is thrown vertically downward with \(5\) m/s. The acceleration of centre of mass of the three-body system will be:
1. \(2\sqrt{2}g\)
2. \(g\)
3. \(\sqrt{2}g\)
4. zero
From a disc of radius \(R,\) a disc of radius \(\frac{R}{2}\) is taken out as shown in the figure. The position of the centre of mass of the remaining disc is on:
1. \({OA}\)
2. \({OB}\)
3. \({OC}\)
4. \({OD}\)
Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration a. The centre of mass has an acceleration:
1. zero
2.
3. a
4. 2a
The angular speed of the wheel of a vehicle is increased from 360 rpm to 1200 rpm in 14 seconds. Its angular acceleration is
\(1.~2\pi ~rad/s^{2}\)
\(2.~28\pi ~rad/s^{2}\)
\(3.~120\pi ~rad/s^{2}\)
\(4.~\pi ~rad/s^{2}\)
Three identical spheres, each of mass M, are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to 2 m (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the center of mass.
1.
2.
3.
4.
A uniform square plate has a small piece \(Q\) of an irregular shape removed and glued to the center of the plate leaving a hole behind in the figure. The \(COM\) of the plate is now in the following quadrant of the \(x\text-y\) plane.
1. \(\text{I}\)
2. \(\text{II}\)
3. \(\text{III}\)
4. \(\text{IV}\)
Three particles of masses \(100~\text{g}\), \(150~\text{g}\), and \(200~\text{g}\) respectively are placed at the vertices of an equilateral triangle of a side \(0.5~\text{m}\) long. What is the position of the centre of mass of three particles?
What is the position of centre of mass of a uniform L-shaped lamina (a thin flat plate) with dimensions as shown? The mass of the lamina is 3 kg.
1. | \(\left(\dfrac{6}{5} , \dfrac{6}{5}\right)\) | 2. | \(\left(\dfrac{1}{5} , \dfrac{6}{5}\right)\) |
3. | \(\left(\dfrac{5}{6} , \dfrac{5}{6}\right)\) | 4. | Can't be determined |
A circular plate of diameter \(d\) is kept in contact with a square plate of edge \(d\) as shown in the figure. The density of the material and the thickness are the same everywhere. The centre of mass of the composite system will be:
1. | inside the circular plate |
2. | inside the square plate |
3. | at the point of contact |
4. | outside the system |
A uniform sphere is placed on a smooth horizontal surface and a horizontal force F is applied on it at a distance h above the surface. The acceleration of the centre
1. is maximum when h = 0
2. is maximum when h = R
3. is maximum when h = 2R
4. is independent of h
If the external forces acting on a system have zero resultant, the centre of mass:
(a) must not move
(b) must not accelerate
(c) may move
(d) may accelerate
Choose the correct options:
1. (a) and (b)
2. (b) and (c)
3. (c) and (d)
4. All of these
The center of mass of a thin conical surface of height \(h\) lies at a distance \(\lambda h\) from its apex, the base of the cone being hollow. The value of \(\lambda\) equals:
1. | \(\dfrac12\) | 2. | \(\dfrac23\) |
3. | \(\dfrac34\) | 4. | \(\dfrac25\) |
The center of mass is defined as \(\vec{R}=\frac{1}{M} \sum_{i} m_{i} \overrightarrow{r_{i}}.\). Suppose we define "center of charge" as \(\vec{R}_{c}=\frac{1}{Q} \sum_{i} q_{i} \vec{r}_{i}\) where qi represents the ith charge placed at \(\vec r_i\) and Q is the total charge of the system. center of charge of a two-charge system
1. May be inside the line segment joining charges
2. May be outside the line segment joining charges
3. May be outside the line segment joining charges
4. Both (1) & (2)
1. | \(1.5~\text{m}\) in the direction of displacement of the man. |
2. | \(0.75~\text{m}\) in the direction of displacement of the man. |
3. | \(1.5~\text{m}\) in the direction opposite to the displacement of the man. |
4. | \(0.75~\text{m}\) in the direction opposite to the displacement of the man. |
1. | \(x_{\Large_B}=A_{0} \sin \omega t\) | 2. | \(x_{\Large_B}=A_{0} \cos \omega t\) |
3. | \(x_{B}=A_{0} \sin (\omega t+\pi)\) | 4. | \(x_{\Large_B}=A_{0} \cos (\omega t+\pi)\) |
1. | \(\dfrac{(2n+1)l}{3}\) | 2. | \(\dfrac{l}{n+1}\) |
3. | \(\dfrac{(n+1)l}{2}\) | 4. | \(\dfrac{2l}{n(n^2+1)}\) |
1. | \(40\) cm from the \(2\) kg particle |
2. | \(60\) cm from the \(3\) kg particle |
3. | \(60\) cm from the \(2\) kg particle |
4. | \(20\) cm from the \(3\) kg particle |
Statement I: | The centre of mass of a system of particles lying on a straight line must lie between the two extreme particles. |
Statement II: | The centre of mass of a system of bodies moving with different velocities, cannot be moving with constant velocity. |
1. | Statement I is incorrect and Statement II is correct. |
2. | Both Statement I and Statement II are correct. |
3. | Both Statement I and Statement II are incorrect. |
4. | Statement I is correct and Statement II is incorrect. |
Assertion (A): | The centre of mass of a proton and an electron, released from their respective positions remains at rest. |
Reason (R): | The centre of mass remains at rest if no external force is applied. |
1. | Both (A) and (R) are True and (R) is the correct explanation of (A). |
2. | Both (A) and (R) are True but (R) is not the correct explanation of (A). |
3. | (A) is True but (R) is False. |
4. | Both (A) and (R) are False. |
Statement I: | The acceleration of the center-of-mass of the system is non-zero. |
Statement II: | The net momentum of the system remains constant. |
1. | Statement I is incorrect and Statement II is correct. |
2. | Both Statement I and Statement II are correct. |
3. | Both Statement I and Statement II are incorrect. |
4. | Statement I is correct and Statement II is incorrect. |
1. | \(\dfrac{m_1}{m_2}\) | 2. | \(\sqrt{\dfrac{m_1}{m_2}}\) |
3. | \(\dfrac{m_2}{m_1}\) | 4. | \(\sqrt{\dfrac{m_2}{m_1}}\) |