Intermediate Mechanics Quiz-II

Q1. A constraint in mechanics is a condition that:

Increases the energy of the system
Restricts the possible motion of the system
Increases the number of particles
Removes all forces

Q2. A particle constrained to move on the surface of a sphere satisfies:

\(x^2+y^2+z^2=R^2\)
\(x+y+z=R^2\)
\(x^2+y^2=R\)
\(xyz=R^2\)

Q3. A constraint that can be expressed in the form \(f(q_1,q_2,\ldots,q_n,t)=0\) is called:

Nonholonomic
Holonomic
Dissipative
Virtual

Q4. A holonomic constraint that does not explicitly depend on time is called:

Rheonomous
Scleronomous
Nonholonomic
Dissipative

Q5. A constraint that explicitly depends on time is called:

Scleronomous
Rheonomous
Conservative
Cyclic

Q6. The minimum number of independent coordinates required to completely describe the configuration of a system is called:

Number of constraints
Number of particles
Number of degrees of freedom
Number of forces

Q7. A free particle moving in three-dimensional space has:

1 degree of freedom
2 degrees of freedom
3 degrees of freedom
6 degrees of freedom

Q8. A particle constrained to move along a straight line has:

1 degree of freedom
2 degrees of freedom
3 degrees of freedom
0 degrees of freedom

Q9. A particle constrained to move on the surface of a sphere has:

1 degree of freedom
2 degrees of freedom
3 degrees of freedom
4 degrees of freedom

Q10. The simple pendulum has:

1 degree of freedom
2 degrees of freedom
3 degrees of freedom
0 degrees of freedom

Q11. Generalized coordinates are usually represented by:

\(F_i\)
\(p_i\)
\(q_i\)
\(E_i\)

Q12. Which of the following can be used as a generalized coordinate?

Distance
Angle
Cartesian coordinate
All of the above

Q13. The space formed by all possible values of the generalized coordinates is called:

Real space
Momentum space
Configuration space
Velocity space

Q14. If a system has \(n\) degrees of freedom, its configuration space has:

One dimension
Three dimensions
\(n\) dimensions
\(2n\) dimensions

Q15. A virtual displacement is:

The actual displacement occurring during a finite time interval
An infinitesimal displacement consistent with the constraints at a fixed time
A displacement caused only by gravity
An impossible displacement

Q16. Virtual displacement is usually represented by:

\(d\vec r\)
\(\delta\vec r\)
\(\dot{\vec r}\)
\(\ddot{\vec r}\)

Q17. According to the principle of virtual work, for a system in equilibrium:

\(\delta W > 0\)
\(\delta W < 0\)
\(\delta W = 0\)
\(\delta W = \infty\)

Q18. The virtual work done by forces \(\vec F_i\) is:

\(\delta W=\sum_i \vec F_i\times\delta\vec r_i\)
\(\delta W=\sum_i \vec F_i\cdot\delta\vec r_i\)
\(\delta W=\sum_i F_i/\delta r_i\)
\(\delta W=\sum_i F_i^2\)

Q19. D'Alembert's principle converts a dynamical problem into a form similar to:

Thermodynamics
Optics
Static equilibrium
Electrostatics only

Q20. D'Alembert's principle may be written as:

\(\sum_i(\vec F_i+m_i\vec a_i)\cdot\delta\vec r_i=0\)
\(\sum_i(\vec F_i-m_i\vec a_i)\cdot\delta\vec r_i=0\)
\(\sum_i m_i\vec a_i=0\)
\(\sum_i\vec F_i=0\) for every moving system

Q21. For a conservative mechanical system, the Lagrangian is:

\(L=T+V\)
\(L=T-V\)
\(L=V-T\)
\(L=TV\)

Q22. In the expression \(L=T-V\), \(T\) represents:

Kinetic energy
Potential energy
Total energy
Torque

Q23. Lagrange's equation for a conservative system is:

\(\frac{d}{dt}\left(\frac{\partial L}{\partial q_i}\right)-\frac{\partial L}{\partial\dot q_i}=0\)
\(\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_i}\right)-\frac{\partial L}{\partial q_i}=0\)
\(\frac{\partial L}{\partial t}=0\)
\(\frac{dL}{dt}=0\)

Q24. The generalized velocity corresponding to \(q_i\) is:

\(q_i^2\)
\(\dot q_i\)
\(\ddot q_i\)
\(1/q_i\)

Q25. The conjugate momentum corresponding to \(q_i\) is defined as:

\(p_i=\frac{\partial L}{\partial q_i}\)
\(p_i=\frac{\partial L}{\partial\dot q_i}\)
\(p_i=mq_i\)
\(p_i=\frac{dL}{dt}\)

Q26. A generalized coordinate that does not explicitly appear in the Lagrangian is called:

Virtual coordinate
Constraint coordinate
Cyclic coordinate
Cartesian coordinate

Q27. If \(q_i\) is cyclic, then:

\(\frac{\partial L}{\partial\dot q_i}=0\)
\(\frac{\partial L}{\partial q_i}=0\)
\(q_i=0\)
\(\dot q_i=0\)

Q28. If \(q_i\) is cyclic, its conjugate momentum is:

Zero
Increasing continuously
Conserved
Always negative

Q29. Conservation of linear momentum is associated with symmetry under:

Spatial translations
Rotations only
Time reversal
Scaling

Q30. Conservation of angular momentum is associated with:

Time-translation symmetry
Rotational symmetry
Spatial-translation symmetry
Reflection only

Q31. Conservation of energy is associated with:

Rotational symmetry
Spatial-translation symmetry
Time-translation symmetry
Mirror symmetry

Q32. Noether's theorem establishes a connection between:

Force and acceleration
Mass and velocity
Continuous symmetries and conservation laws
Temperature and energy

Q33. If the laws of physics are unchanged when the origin of time is shifted, the corresponding conserved quantity is:

Linear momentum
Angular momentum
Energy
Position

Q34. The Lagrangian of a one-dimensional harmonic oscillator is:

\(L=\frac12m\dot{x}^2+\frac12kx^2\)
\(L=\frac12m\dot{x}^2-\frac12kx^2\)
\(L=m\dot{x}-kx\)
\(L=\frac12kx^2-\frac12m\dot{x}^2\)

Q35. Applying Lagrange's equation to the harmonic oscillator gives:

\(m\ddot{x}-kx=0\)
\(m\ddot{x}+kx=0\)
\(m\dot{x}+kx=0\)
\(m\ddot{x}=0\)

Q36. The natural angular frequency of a linear harmonic oscillator is:

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

Q37. A convenient generalized coordinate for a simple pendulum is:

Its mass \(m\)
Its length \(l\)
Its angular displacement \(\theta\)
Gravitational acceleration \(g\)

Q38. The kinetic energy of a simple pendulum of length \(l\) is:

\(\frac12m\dot{\theta}^2\)
\(\frac12ml^2\dot{\theta}^2\)
\(mgl\cos\theta\)
\(\frac12ml\dot{\theta}\)

Q39. Taking the lowest point as zero potential energy, the potential energy of a simple pendulum is:

\(mgl\cos\theta\)
\(-mgl\sin\theta\)
\(mgl(1-\cos\theta)\)
\(mgl\theta\)

Q40. The Lagrangian of a simple pendulum can be written as:

\(L=\frac12ml^2\dot{\theta}^2-mgl(1-\cos\theta)\)
\(L=\frac12ml^2\dot{\theta}^2+mgl(1-\cos\theta)\)
\(L=mgl\dot{\theta}\)
\(L=\frac12m\dot{\theta}^2\)

Q41. Lagrange's equation gives the exact equation of motion of a simple pendulum as:

\(\ddot{\theta}-\frac{g}{l}\sin\theta=0\)
\(\ddot{\theta}+\frac{g}{l}\sin\theta=0\)
\(\ddot{\theta}+\frac{g}{l}\cos\theta=0\)
\(\dot{\theta}+\frac{g}{l}\theta=0\)

Q42. For small angular displacements, we use the approximation:

\(\sin\theta\approx1\)
\(\sin\theta\approx\theta^2\)
\(\sin\theta\approx\theta\)
\(\sin\theta\approx0\) for every \(\theta\)

Q43. Under the small-angle approximation, the pendulum equation becomes:

\(\ddot{\theta}+\frac{g}{l}\theta=0\)
\(\ddot{\theta}-\frac{g}{l}\theta=0\)
\(\dot{\theta}+\frac{g}{l}\theta=0\)
\(\ddot{\theta}=g\)

Q44. The angular frequency of small oscillations of a simple pendulum is:

\(\sqrt{l/g}\)
\(\sqrt{g/l}\)
\(g/l\)
\(2\pi\sqrt{g/l}\)

Q45. The period of small oscillations of a simple pendulum is:

\(2\pi\sqrt{g/l}\)
\(2\pi\sqrt{l/g}\)
\(\sqrt{l/g}\)
\(2\pi l/g\)

Q46. One major advantage of the Lagrangian formulation is that:

It can only use Cartesian coordinates
It avoids kinetic energy
It can conveniently use generalized coordinates suited to the constraints
It applies only to one-dimensional systems

Q47. Newtonian mechanics primarily describes motion using:

Only potential energy
Forces and accelerations
Only generalized momentum
Only symmetry

Q48. The Lagrangian approach is particularly useful for systems containing:

No motion
No energy
Constraints
Only constant forces

Q49. In many constrained systems, the Lagrangian method has the advantage that:

All constraint forces must always be calculated first
Suitable generalized coordinates can eliminate the need to explicitly calculate some constraint forces
Newton's laws become incorrect
Potential energy must always be zero

Q50. Which statement correctly compares Newtonian and Lagrangian mechanics?

Newtonian mechanics is correct but Lagrangian mechanics is only approximate
Lagrangian mechanics applies only to oscillations
Newtonian mechanics cannot describe constrained systems
They are equivalent formulations of classical mechanics, but the Lagrangian method can be more convenient for constrained systems

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