The density of a rod having length \(l\) varies as \(\rho=c+dx,\) where \(x\) is the distance from the left end. The distance from origin to the centre of mass is:

1. \(\frac{3cl+2Dl^2}{3(2c+Dl)}\)
2. \(\frac{3cl+4Dl^2}{3(2c+Dl)}\)
3. \(\frac{2cl+3Dl^2}{3(2c+1)}\)
4. \(\frac{cl+3Dl^2}{3(2c+Dl)}\)

Subtopic:  Center of Mass |
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One end of a massless spring of constant 100 N/m and natural length 0.5 m is fixed and the other end is connected to a particle of mass 0.5 kg lying on a frictionless horizontal table. The spring remains horizontal. If the mass is made to rotate at an angular velocity of 2 rad/s, then the elongation of the spring is:

1.  0.1 m
2.  10 cm
3.  1 cm
4.  0.01 cm

Subtopic:  Uniform Circular Motion |
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A block slides down on an incline of angle 30° with an acceleration \(\frac g4.\) Find the kinetic friction coefficient.
1. \(\frac{1}{2\sqrt2}\)
2. \(0.6\)
3. \(\frac{1}{2\sqrt3}\)
4. \(\frac{1}{\sqrt2}\)

Subtopic:  Friction |
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Two long straight wires, each carrying an electric current of 5 A, are kept parallel to each other at a separation of 2.5 cm. Find the magnitude of the magnetic force experiment by 10 cm of a wire.

1.  4.0 × 10-4 N

2.  3.5 × 10-6 N

3.  2.0 × 10-5 N

4.  2.0 × 10-9 N

Subtopic:  Force between Current Carrying Wires |
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A wire of resistance 10 Q is bent to form a complete circle. Find its resistance between two diametrically opposite points. 
   
1. \(5~\Omega\)
2. \(2.5~\Omega\)
3. \(1.25~\Omega\)
4. \(\frac{10}{3}~\Omega\)

Subtopic:  Combination of Resistors |
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Find the resistance of a hollow cylindrical Conductor with the inner and outer radii of length 1.0 mm and 2.0 mm respectively. The resistivity of the material is 2.0 x 10-8 \(\Omega\) m. The length of the conductor is 10 m.

1.  2.1 x 10-3 \(\Omega\)

2.  1.3 x 10-4 \(\Omega\)

3.  3.2 x 10-4 \(\Omega\)

4.  4.6 x 10-2 \(\Omega\)

Subtopic:  Derivation of Ohm's Law |
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Three equal charges, each having a magnitude of 2.0 × 10-6 C, are placed at the three corners of a right-angled triangle of sides 3 cm, 4 cm and 5 cm. The force (in magnitude) on the charge at the right-angled corner is:

1.  50 N
2.  26 N
3.  29 N
4.  45.9 N

Subtopic:  Coulomb's Law |
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A diatomic gas (\(\gamma=1.4\)) does \(200~\mathrm{J}\) of work when it is expanded isobarically. The heat given to the gas in the process is:
1. \( 500 \mathrm{~J} \)
2. \( 700 \mathrm{~J}\)

3. \( 600 \mathrm{~J}\)
4. \( 900 \mathrm{~J} \)
 

Subtopic:  Work Done by a Gas |
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A uniform ring of mass \(m\) and radius a is placed directly above a uniform sphere of mass m and of equal to radius. The centre of the ring is at a distance \(\sqrt3a\) from the centre of the sphere. The gravitational force (F) exerted by the sphere on the ring is:
1. \(\frac{3G~Mm}{8a^2}\)
2. \(\frac{2G~Mm}{3a^2}\)
3. \(\frac{7G~Mm}{\sqrt2a^2}\)
4. \(\frac{3G~Mm}{a^2}\)

Subtopic:  Gravitational Field |
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A projectile is fired with a velocity \(u\) at angle \(\theta\) with the ground surface. During the motion at any time, it is making an angle \(\alpha\) with the ground surface. The speed of the particle at this time will be:
1. \(u\cos\theta~\sec\alpha\)
2. \(u\cos\theta.\tan\alpha\)
3. \(u^2\cos^2\theta.\sin\alpha\)
4. \(u\sin\theta.\sin\alpha\)

Subtopic:  Projectile Motion |
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