Center OF Mass flash cards

1. 1.Define the centre of mass (CM) of a system of particles.

   The CM is a point that moves as though the total mass of the system were concentrated there and all external forces acted at that point. Its position is the mass-weighted average of all particle positions: $\vec r_{cm}=\dfrac{\sum m_i\vec r_i}{\sum m_i}$.

   Hint: Mass-weighted average position.

2. 2.Write the position vector of the CM for a system of $n$ discrete particles.

   $\vec r_{cm}=\dfrac{m_1\vec r_1+m_2\vec r_2+\cdots+m_n\vec r_n}{m_1+m_2+\cdots+m_n}=\dfrac{1}{M}\sum_i m_i\vec r_i$, where $M=\sum m_i$ is the total mass.

   Hint: Sum of $m_i\vec r_i$ over total mass.

3. 3.For a continuous body, how is the CM position found?

   Replace the sum by an integral: $\vec r_{cm}=\dfrac{1}{M}\displaystyle\int \vec r\,dm$, where $dm$ is a mass element and $M=\int dm$.

   Hint: Integrate $\vec r\,dm$, divide by $M$.

4. 4.For two particles of masses $m_1$ and $m_2$ separated by distance $d$, where does the CM lie?

   On the line joining them, at distance $\dfrac{m_2 d}{m_1+m_2}$ from $m_1$ (and $\dfrac{m_1 d}{m_1+m_2}$ from $m_2$). It lies closer to the heavier mass.

   Hint: Distances inversely proportional to masses.

5. 5.Does the CM of a body always lie inside the material of the body?

   No. The CM can lie outside the material — e.g. a ring, a hollow sphere, or an L-shaped/horseshoe object has its CM in empty space.

   Hint: Think of a ring.

6. 6.How does symmetry help locate the CM of a uniform body?

   For a uniform (homogeneous) body, the CM lies on every axis/plane/point of symmetry. If the body has a centre of symmetry, the CM is there.

   Hint: CM sits on symmetry elements.

7. 7.Distinguish centre of mass from centre of gravity.

   CM depends only on mass distribution. Centre of gravity is where total gravitational torque is zero. In a uniform gravitational field ($g$ constant) they coincide; they differ only for very large bodies where $g$ varies.

   Hint: They coincide when $g$ is uniform.

8. 8.Where is the CM of a uniform straight rod of length $L$?

   At its geometric midpoint, i.e. at $L/2$ from either end.

   Hint: Midpoint.

9. 9.Where is the CM of a uniform semicircular ring (wire) of radius $R$?

   On the axis of symmetry, at a distance $\dfrac{2R}{\pi}$ from the centre.

   Hint: $2R/\pi$.

10. 10.Where is the CM of a uniform semicircular disc (lamina) of radius $R$?

    On the symmetry axis at distance $\dfrac{4R}{3\pi}$ from the centre.

    Hint: $4R/3\pi$.

11. 11.Where is the CM of a uniform solid hemisphere of radius $R$?

    On the symmetry axis at distance $\dfrac{3R}{8}$ from the flat face's centre.

    Hint: $3R/8$.

12. 12.Where is the CM of a uniform hollow hemisphere (hemispherical shell) of radius $R$?

    On the axis at distance $\dfrac{R}{2}$ from the centre of the flat face.

    Hint: $R/2$.

13. 13.Where is the CM of a uniform solid cone of height $h$?

    On the axis, at height $\dfrac{h}{4}$ from the base (i.e. $\tfrac34 h$ from the apex).

    Hint: $h/4$ from base.

14. 14.Where is the CM of a uniform triangular lamina?

    At its centroid — the intersection of the medians, located at $1/3$ of each median's length from the corresponding side.

    Hint: Centroid = average of vertices.

15. 15.Where is the CM of a thin uniform circular arc of radius $R$ subtending angle $2\alpha$ at the centre?

    On the bisector of the arc, at distance $\dfrac{R\sin\alpha}{\alpha}$ from the centre (angle in radians).

    Hint: $R\sin\alpha/\alpha$; check semicircle limit.

16. 16.How do you handle the CM of a body with a cavity (hole cut out)?

    Treat the removed part as negative mass. $\vec r_{cm}=\dfrac{M\vec r_{full}-m\vec r_{hole}}{M-m}$, where $M$ is the full body and $m$ the removed piece.

    Hint: Superposition with negative mass.

17. 17.A uniform disc of radius $R$ has a circular hole of radius $R/2$ cut, the hole's edge touching the disc's edge. Where is the new CM?

    Along the line joining the centres, shifted away from the hole by $\dfrac{R}{6}$ from the big disc's centre. (Removed mass $=M/4$; hole centre at $R/2$: shift $=\frac{(M/4)(R/2)}{M-M/4}=R/6$.)

    Hint: Negative mass $M/4$ at $R/2$.

18. 18.How is the velocity of the CM defined?

    $\vec v_{cm}=\dfrac{d\vec r_{cm}}{dt}=\dfrac{1}{M}\sum_i m_i\vec v_i=\dfrac{\vec P_{total}}{M}$, i.e. total momentum divided by total mass.

    Hint: $\vec P=M\vec v_{cm}$.

19. 19.State the equation of motion of the CM.

    $M\vec a_{cm}=\vec F_{ext}$: the CM accelerates as if all mass were concentrated there and the net external force acted on it. Internal forces cancel in pairs (Newton's third law).

    Hint: Only external forces move the CM.

20. 20.Why do internal forces not affect the motion of the CM?

    By Newton's third law internal forces occur in equal and opposite pairs, so their vector sum is zero and they contribute nothing to $M\vec a_{cm}$.

    Hint: Action–reaction cancels.

21. 21.A shell fired on a parabolic path explodes mid-air. What path does the CM follow?

    The CM continues on the original parabolic trajectory, because the explosion is due to internal forces only; gravity (external) is unchanged.

    Hint: Internal explosion doesn't shift the CM path.

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