Central-force Motion and Oscillations - Classical Mechanics - Physics Previous Year Questions

1

2007 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2007

A space station moving in a circular orbit around the Earth goes into a new bound orbit by firing its engine radially outwards. This orbit is

A

a larger circle

B

a smaller circle

C

an ellipse

D

a parabola

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2

2008 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2008

A mass \( m \) is constrained to move on a horizontal frictionless surface. It is set in circular motion with radius \( r_0 \) and angular speed \( \omega_0 \) by an applied force \( \vec{F} \) communicated through an inextensible thread that passes through a hole on the surface as shown in the figure. This force is then suddenly doubled. The magnitude of the radial velocity of the mass

A

increases till the mass falls into the hole.

B

decreases till the mass falls into the hole.

C

remains constant.

D

becomes zero at a radius \(r_1 \) where \(0 < r_1 < r_0 \).

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3

2008 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2008

An inelastic ball of mass \( m \) has been thrown vertically upwards from the ground at \( z=0 \). The initial kinetic energy of the ball is \( E \). The phase trajectory of the ball after successive bouncing on the ground is

A

graph with p_z vs z showing a semicircle in the right half plane with center at origin

B

graph with p_z vs z showing multiple nested semicircles in the right half plane with center at origin

C

graph with p_z vs z showing a single closed loop in the right half plane, with arrows indicating clockwise motion

D

graph with p_z vs z showing two symmetric closed loops in the right and left half planes, with arrows indicating counter-clockwise motion on the right and clockwise on the left

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4

2009 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2009

The Lagrangian of a diatomic molecule is given by \(L = \frac{m}{2}(\dot{x}_1^2 + \dot{x}_2^2) - \frac{k}{2}x_1x_2\), where \(m\) is the mass of each of the atoms and \(x_1\) and \(x_2\) are the displacements of atoms measured from the equilibrium position and \(k > 0\). The normal frequencies are

A

\(\pm\left(\frac{k}{m}\right)^{1/2}\)

B

\(\pm\left(\frac{k}{m}\right)^{1/4}\)

C

\(\pm\left(\frac{k}{2m}\right)^{1/4}\)

D

\(\pm\left(\frac{k}{2m}\right)^{1/2}\)

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5

2010 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2010

A particle is placed in a region with the potential \( V(x) = \frac{1}{2} k x^2 - \frac{\lambda}{3} x^3 \), where \( k, \lambda > 0 \). Then,

A

\(x = 0 \) and \(x = \frac{k}{\lambda} \) are points of stable equilibrium

B

\(x = 0 \) is a point of stable equilibrium and \(x = \frac{k}{\lambda} \) is a point of unstable equilibrium

C

\(x = 0 \) and \(x = \frac{k}{\lambda} \) are points of unstable equilibrium

D

There are no points of stable or unstable equilibrium

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6

2011 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2011

If a force \( \vec{F} \) is derivable from a potential function \( V(r) \), where \( r \) is the distance from the origin of the coordinate system, it follows that

A

\(\vec{\nabla} \times \vec{F} = 0 \)

B

\(\vec{\nabla} \cdot \vec{F} = 0 \)

C

\(\vec{\nabla} V = 0 \)

D

\(\nabla^2 V = 0 \)

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7

2011 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2011

If a force F⃗ is derivable from a potential function V(r), where r is the distance from the origin of the coordinate system, it follows that

A

∇⃗×F⃗ = 0

B

∇⃗·F⃗ = 0

C

∇⃗V = 0

D

∇²V = 0

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8

2011 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2011

Two bodies of mass \( m \) and \( 2m \) are connected by a spring of spring constant \( k \). The frequency of the normal mode is

A

\(\sqrt{3k/2m} \)

B

\(\sqrt{k/m} \)

C

\(\sqrt{2k/3m} \)

D

\(\sqrt{k/2m} \)

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9

2012 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2012

In a central force field, the trajectory of a particle of mass $m$ and angular momentum $L$ in plane polar coordinates is given by, \(\frac{1}{r} = \frac{m}{L^2} (1 + e \cos \theta)\) where, $e$ is the eccentricity of the particle's motion. Which one of the following choices for $e$ gives rise to a parabolic trajectory?

A

e = 0

B

e = 1

C

0 < e < 1

D

e > 1

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10

2012 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2012

A particle of unit mass moves along the \(x\)-axis under the influence of a potential, \(V(x)=x(x-2)^2\). The particle is found to be in stable equilibrium at the point \(x=2\). The time period of oscillation of the particle is

A

\(\frac{\pi}{2}\)

B

\(\pi\)

C

\(\frac{3\pi}{2}\)

D

\(2\pi\)

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11

2013 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2013

Consider two small blocks, each of mass \( M \), attached to two identical springs. One of the springs is attached to the wall, as shown in the figure. The spring constant of each spring is \( k \). The masses slide along the surface and the friction is negligible. The frequency of one of the normal modes of the system is,

A

\(\frac{\sqrt{3+\sqrt{2}}}{2} \sqrt{\frac{k}{M}} \)

B

\(\frac{\sqrt{3+\sqrt{2}}}{2} \sqrt{\frac{k}{M}} \)

C

\(\frac{\sqrt{3+\sqrt{5}}}{2} \sqrt{\frac{k}{M}} \)

D

\(\frac{\sqrt{3+\sqrt{5}}}{2} \sqrt{\frac{k}{M}} \)

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12

2014 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2014

Two masses \(m\) and \(3m\) are attached to the two ends of a massless spring with force constant \(K\). If \(m = 100\) g and \(K = 0.3\) N/m, then the natural angular frequency of oscillation is _____________ Hz.

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13

2014 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2014

A particle of mass m is in a potential given by \[V(r) = -\frac{a}{r} + \frac{a r_0^2}{3 r^3},\] where a and r_0 are positive constants. When disturbed slightly from its stable equilibrium position, it undergoes a simple harmonic oscillation. The time period of oscillation is

A

\(2\pi \sqrt{\frac{m r_0^3}{2a}}\)

B

\(2\pi \sqrt{\frac{m r_0^3}{a}}\)

C

\(2\pi \sqrt{\frac{2 m r_0^3}{a}}\)

D

\(4\pi \sqrt{\frac{m r_0^3}{a}}\)

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14

2014 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2014

A planet of mass \(m\) moves in a circular orbit of radius \(r_0\) in the gravitational potential \(V(r) = -\frac{k}{r}\), where \(k\) is a positive constant. The orbital angular momentum of the planet is

A

\(2r_0 km\)

B

\(\sqrt{2r_0 km}\)

C

\(r_0 km\)

D

\(\sqrt{r_0 km}\)

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15

2015 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2015

A satellite is moving in a circular orbit around the Earth. If T, V and E are its average kinetic, average potential and total energies, respectively, then which one of the following options is correct?

A

\(V = -2T; E = -T\)

B

\(V = -T; E = 0\)

C

\(V = -T/2; E = T/2\)

D

\(V = -3T/2; E = -T/2\)

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16

2015 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2015

Four forces are given below in Cartesian and spherical polar coordinates.
(i) \( \vec{F}_1 = K \exp(-r^2 / R^2) \hat{y} \)
(ii) \( \vec{F}_2 = K (x^3 \hat{y} - y^3 \hat{x}) \)
(iii) \( \vec{F}_3 = K (x^2 \hat{x} + y^2 \hat{y}) \)
(iv) \( \vec{F}_4 = K (\hat{\phi} / r) \)
where \( K \) is a constant. Identify the correct option.

A

(iii) and (iv) are conservative but (i) and (ii) are not

B

(i) and (ii) are conservative but (iii) and (iv) are not

C

(ii) and (iii) are conservative but (i) and (iv) are not

D

(i) and (iii) are conservative but (ii) and (iv) are not

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17

2016 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2016

Two blocks are connected by a spring of spring constant k. One block has mass m and the other block has mass 2m. If the ratio k/m = 4 s^{-2}, the angular frequency of vibration ω of the two block spring system in s^{-1} is ______. (Give your answer upto two decimal places)

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18

2016 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2016

A particle moving under the influence of a central force \( \vec{F}(\vec{r}) = -k \vec{r} \) (where \( \vec{r} \) is the position vector of the particle and k is a positive constant) has non-zero angular momentum. Which of the following curves is a possible orbit for this particle?

A

A straight line segment passing through the origin.

B

An ellipse with its center at the origin.

C

An ellipse with one of the foci at the origin.

D

A parabola with its vertex at the origin.

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19

2017 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2017

Two identical masses of 10 gm each are connected by a massless spring of spring constant 1 N/m. The non-zero angular eigenfrequency of the system is ______ rad/s. (up to two decimal places.)

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20

2017 · Physics · Classical Mechanics · Central-force Motion and Oscillations

Physics (PH) 2017

The phase space trajectory of an otherwise free particle bouncing between two hard walls elastically in one dimension is a

A

straight line

B

parabola

C

rectangle

D

circle

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