Centre of Mass Definition and Position - UNSOLVED PRACTICE SET
Chapter: System of Particles and Rotational Motion | Topic: Centre of Mass Definition and Position
CENTRE OF MASS DEFINITION AND POSITION - UNSOLVED PRACTICE SET
Topic: Centre of Mass Definition and Position
Multiple Choice Questions
Q1. The centre of mass of a system of particles is the point where:
- The total mass of the system is concentrated
- The total external force is zero
- The entire mass of the system can be assumed to be concentrated for translational motion
- The gravitational potential is maximum
Q2. For a system of two particles of masses m₁ and m₂ separated by distance d, the centre of mass lies:
- At the midpoint regardless of masses
- Closer to the heavier mass
- Closer to the lighter mass
- Outside the line joining the two particles
Q3. The position vector of the centre of mass of a system of n particles is given by:
- R_cm = Σmᵢrᵢ
- R_cm = (Σmᵢrᵢ)/Σmᵢ
- R_cm = (Σmᵢrᵢ)/n
- R_cm = Σrᵢ/n
Q4. If the origin is shifted to the centre of mass of a system, then:
- The total mass becomes zero
- The sum of mᵢrᵢ becomes zero
- The total momentum becomes zero
- Both (b) and (c)
Q5. The centre of mass of a uniform rod of length L lies at:
- One end
- L/4 from one end
- L/2 from one end
- L/3 from one end
Q6. When a cricketer throws a baton during a march-past in your school sports day, the centre of mass of the baton follows:
- A complicated curved path
- A parabolic path
- A straight line path
- A circular path
Short Answer Questions
Q7. Define centre of mass. Is it necessary for the centre of mass to lie within the body? Give one example where it does not.
Q8. Two particles of masses 2 kg and 4 kg are placed at x = 0 and x = 6 m respectively on the x-axis. Find the position of their centre of mass.
Q9. Explain why the centre of mass of a uniform triangular lamina lies at its centroid.
Q10. Three particles of equal mass are placed at the vertices of an equilateral triangle. Where does the centre of mass lie?
Q11. A uniform disc of radius R has a hole of radius R/2 cut out from it, with the centre of the hole at R/2 from the centre of the disc. Explain how you would find the centre of mass of the remaining portion.
Q12. Does the centre of mass of a body change if its shape changes but mass remains the same? Explain with an example.
Long Answer Questions
Q13. Define centre of mass for a system of discrete particles. Derive the expression for the position vector of the centre of mass:
R_cm = (m₁r₁ + m₂r₂ + ... + mₙrₙ)/(m₁ + m₂ + ... + mₙ)
Extend this to a continuous body and write the integral form. Explain the significance of the centre of mass in describing the translational motion of a system.
Q14. A uniform rod of length L and mass M has a point mass m attached to one end.
(i) Find the position of the centre of mass of the system from the other end of the rod.
(ii) If the rod is pivoted at its centre, will the system be in equilibrium? Explain.
(iii) Where should the pivot be placed so that the system balances horizontally?
(iv) How does the position of the centre of mass change if the point mass is moved to the centre of the rod?
Q15. A student cuts a uniform square sheet of side 2a into four equal smaller squares. She removes one small square from a corner.
(i) Find the coordinates of the centre of mass of the remaining L-shaped figure, taking the corner where the square was removed as origin.
(ii) Verify that the centre of mass lies outside the material of the body.
(iii) If the removed square is reattached at the diagonally opposite corner, where is the new centre of mass?
(iv) Explain why knowing the centre of mass is useful in practical applications like designing vehicles and aircraft.
Numerical / Application-Based Problems
Q16. Four particles of masses 1 kg, 2 kg, 3 kg, and 4 kg are placed at the corners of a square of side 1 m. The coordinates are: (0,0), (1,0), (1,1), and (0,1) respectively.
(i) Find the coordinates of the centre of mass.
(ii) If the 4 kg mass is moved to (2,2), find the new centre of mass.
(iii) A fifth particle of mass 5 kg is placed at the centre of mass of the original four-particle system. Where is the new centre of mass?
(iv) If all four masses are replaced by a single particle at the centre of mass, what mass should it have to produce the same gravitational effect at a distant point?
Q17. A uniform rod AB of length 1.2 m and mass 3 kg has two point masses attached: 2 kg at A and 5 kg at B.
(i) Find the position of the centre of mass from end A.
(ii) The rod is placed on a knife-edge. Where should the knife-edge be placed for balance?
(iii) If the 5 kg mass is moved to the centre of the rod, find the new centre of mass.
(iv) The system is now suspended from the new centre of mass. Will it remain horizontal? Explain.
Q18. In a school physics project, a student builds a mobile art piece using three uniform rods and three decorative objects. Rod 1 is 0.6 m long, mass 0.4 kg, with a 0.3 kg object at one end. Rod 2 is 0.5 m long, mass 0.3 kg, with a 0.2 kg object at one end. Rod 3 is 0.4 m long, mass 0.2 kg, with a 0.1 kg object at one end. The rods are connected as follows: Rod 1 is suspended from its centre, Rod 2 is attached to one end of Rod 1, and Rod 3 is attached to the free end of Rod 2.
(i) Find the centre of mass of each rod-object combination.
(ii) Find the overall centre of mass of the entire mobile.
(iii) Where should the top suspension point be placed for the mobile to hang in equilibrium?
(iv) If the student adds a 0.5 kg object at the free end of Rod 3, how does the suspension point change?