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Q. No.A ball experiences an angular acceleration given by:
\(\alpha=(6 {t}^2-2 {t}),\)
where \(t\) is in seconds.
At \(t=0,\) the ball has an angular velocity of \(10\) rad/s and an angular position of \(4\) rad. Which of the following expressions correctly represents the angular position \(\theta({t})\) of the ball?
1.
\( \dfrac{3}{2} t^4-t^2+10 t \)
2.
\(\dfrac{t^4}{2}-\dfrac{t^3}{3}+10 t+4 \)
3.
\( \dfrac{2 t^4}{3}-\dfrac{t^3}{6}+10 t+12 \)
4.
\( 2 t^4-\dfrac{t^3}{2}+5 t+4 \)
Subtopic: Rotational Motion: Kinematics |
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A \(\sqrt {34}\) m long ladder weighing \(10\) kg leans on a frictionless wall. Its feet rest on the floor \(3\) m away from the wall as shown in the figure. If \(\mathrm F_ \mathrm f\) and \(\mathrm F_ \mathrm w\) are the reaction forces of the floor and the wall, then ratio of \(\mathrm F_ \mathrm w / \mathrm F_ \mathrm f\) will be: (Take \(g=10\) m/s2)
1. \(\dfrac{6}{\sqrt{110}} \)
2. \(\dfrac{3}{\sqrt{113}} \)
3. \(\dfrac{3}{\sqrt{109}} \)
4. \( \dfrac{2}{\sqrt{109}}\)
1. \(\dfrac{6}{\sqrt{110}} \)
2. \(\dfrac{3}{\sqrt{113}} \)
3. \(\dfrac{3}{\sqrt{109}} \)
4. \( \dfrac{2}{\sqrt{109}}\)
Subtopic: Torque |
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A uniform disc with mass \(M=4\) kg and radius \(R=10\) cm is mounted on a fixed horizontal axle as shown in the figure. A block with mass \(m=2\) kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is: (Take \(g=10\) ms–2)
1. \(12\) N
2. \(20\) N
3. \(10\) N
4. \(15\) N
1. \(12\) N
2. \(20\) N
3. \(10\) N
4. \(15\) N
Subtopic: Rotational Motion: Dynamics |
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The position vector of a \(1\) kg object is \(\vec r = (3 \hat i +\hat j )~ \text {m}\) and its velocity vector is \(\vec{v}=(3 \hat{\mathrm{j}}+\hat{k}) ~\text{ms}^{-1}.\) If the magnitude of its angular momentum is \(\sqrt x ~\text {N-ms},\) then the value of \(x \) will be:
| 1. | \(67\) | 2. | \(91\) |
| 3. | \(43\) | 4. | \(66\) |
Subtopic: Angular Momentum |
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Match List-I with List-II:
Codes:
| List-I | List-II | ||
| (A) | Moment of inertia of a solid sphere of radius \(R\) about any tangent | (I) | \({ 5 \over 3}MR^2\) |
| (B) | Moment of inertia of a hollow sphere of radius \(R\) about any tangent | (II) | \({ 7 \over 5} MR^2\) |
| (C) | Moment of inertia of a circular ring of radius \(R\) about its diameter | (III) | \({ 1 \over 4} MR^2\) |
| (D) | Moment of inertia of a circular disk of radius \(R\) about any diameter | (IV) | \({ 1 \over 2} MR^2\) |
Codes:
| 1. | A-II, B-I, C-IV, D-III |
| 2. | A-I, B-II, C-IV, D-III |
| 3. | A-II, B-I, C-III, D-IV |
| 4. | A-I, B-II, C-III, D-IV |
Subtopic: Moment of Inertia |
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Two blocks of masses \(10~\text{kg}\) and \(30~\text{kg}\) are placed on the same straight line with coordinates \((0,0)~\text{cm}\) and \((x,0)~\text{cm}\) respectively. The block of \(10~\text{kg}\) is moved on the same line through a distance of \(6~\text{cm}\) towards the other block. The distance through which the block of \(30~\text{kg}\) must be moved to keep the position of centre of mass of the system unchanged is:
| 1. | \(4~\text{cm}\) towards the \(10~\text{kg}\) block |
| 2. | \(2~\text{cm}\) away from the \(10~\text{kg}\) block |
| 3. | \(2~\text{cm}\) towards the \(10~\text{kg}\) block |
| 4. | \(4~\text{cm}\) away from the \(10~\text{kg}\) block |
Subtopic: Center of Mass |
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A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of the rotational kinetic energy of the ball to its total kinetic energy is:
1. \(2\over 5\)
2. \(2\over 7\)
3. \(1\over 5\)
4. \(7\over 10\)
1. \(2\over 5\)
2. \(2\over 7\)
3. \(1\over 5\)
4. \(7\over 10\)
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A thin circular ring of mass \(M\) and radius \(R\) is rotating with a constant angular velocity of \(2~\text{rad/sec}\) in a horizontal plane about an axis vertical to its plane and passing through the centre of the ring. If two objects each of mass \(m\) be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity of:
| 1. | \(\dfrac{M}{M+m}~\text{rad/s}\) | 2. | \(\dfrac{M+2m}{2M}~\text{rad/s}\) |
| 3. | \(\dfrac{2M}{M+2m}~\text{rad/s}\) | 4. | \(\dfrac{2(M+2m)}{M}~\text{rad/s}\) |
Subtopic: Angular Momentum |
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The moment of Inertia (M.I.) of four bodies having the same mass \(M\) and radius \(2R\) are as follows:
If \(2(I_2+I_3)+I_4=xI_1,\) then the value of \(x\) will be:
1. \(3\)
2. \(5\)
3. \(7\)
4. \(9\)
| 1. | \(I_1=\) M.I. of a solid sphere about its diameter |
| 2. | \(I_2=\)M.I. of solid cylinder about its axis |
| 3. | \(I_3=\)M.I. of a solid circular disc about its diameter |
| 4. | \(I_4=\) M.I. of a thin circular ring about its diameter |
If \(2(I_2+I_3)+I_4=xI_1,\) then the value of \(x\) will be:
1. \(3\)
2. \(5\)
3. \(7\)
4. \(9\)
Subtopic: Moment of Inertia |
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A force \(\vec{{F}}=3 \hat{i}+4 \hat{j}-2 \hat{k}\) acts on a particle located at a position vector \(\vec{r}=2 \hat{i}+ \hat{j}+2 \hat{k}.\) What is the torque of this force about the origin?
1. \(3 \hat{i}+4 \hat{j}-2 \hat{k} \)
2. \(-10 \hat{i}+10 \hat{j}+5 \hat{k} \)
3. \(10 \hat{i}+5 \hat{j}-10 \hat{k} \)
4. \(10 \hat{i}+\hat{j}-5 \hat{k}\)
1. \(3 \hat{i}+4 \hat{j}-2 \hat{k} \)
2. \(-10 \hat{i}+10 \hat{j}+5 \hat{k} \)
3. \(10 \hat{i}+5 \hat{j}-10 \hat{k} \)
4. \(10 \hat{i}+\hat{j}-5 \hat{k}\)
Subtopic: Torque |
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