Rotation flash cards

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

   The COM is the point where the entire mass of the system may be assumed to be concentrated for describing its translational motion. For discrete particles $\vec{R}_{cm}=\dfrac{\sum m_i\,\vec{r}_i}{\sum m_i}$.

   Hint: Mass-weighted average of positions.

2. 2.Give the x-coordinate of the COM for a two-particle system of masses $m_1,m_2$ at $x_1,x_2$.

   $x_{cm}=\dfrac{m_1 x_1+m_2 x_2}{m_1+m_2}$. It lies on the line joining them, closer to the heavier mass.

   Hint: Weighted mean along the line joining.

3. 3.For a continuous body, write the COM position vector integral.

   $\vec{R}_{cm}=\dfrac{1}{M}\int \vec{r}\,dm$, where $M$ is the total mass and the integral runs over the whole body.

   Hint: Replace sum by integral over $dm$.

4. 4.Where does the COM of a uniform symmetric body lie?

   At its geometric centre (centre of symmetry) — e.g. centre of a uniform rod, ring, disc, sphere, or rectangular plate.

   Hint: Symmetry fixes the point.

5. 5.Does the COM always lie inside the material of the body?

   No. For bodies like a ring, a hollow sphere, or an L-shaped lamina the COM can lie in empty space (e.g. the centre of a ring has no material).

   Hint: Ring centre example.

6. 6.How does the COM of an isolated system move when internal forces act?

   Internal forces cancel in pairs (Newton's third law), so they cannot change the COM motion. With zero external force the COM moves with constant velocity (or stays at rest).

   Hint: Internal forces sum to zero.

7. 7.Write Newton's second law for a system in terms of its COM.

   $\vec{F}_{ext}=M\,\vec{a}_{cm}$, where $M$ is total mass and $\vec{a}_{cm}$ is the acceleration of the COM. Only external forces matter.

   Hint: Total external force drives the COM.

8. 8.Define the total linear momentum of a system in terms of COM.

   $\vec{P}=M\,\vec{v}_{cm}$. The total momentum equals total mass times COM velocity.

   Hint: $\vec P$ tracks the COM.

9. 9.State the principle of conservation of linear momentum for a system.

   If the net external force is zero, $\vec{P}=M\,\vec{v}_{cm}$ is constant, so $\vec{v}_{cm}$ is constant. Internal interactions (explosions, collisions) do not change $\vec P$.

   Hint: $\vec F_{ext}=0 \Rightarrow \vec P$ constant.

10. 10.A shell moving in a parabolic path explodes in mid-air. What path does its COM follow?

    The COM continues along the same original parabolic trajectory, since the explosion is due to internal forces only; gravity (external) is unchanged.

    Hint: Explosion = internal force.

11. 11.Define angular displacement, angular velocity, and angular acceleration.

    Angular displacement $\theta$ (rad); angular velocity $\omega=\dfrac{d\theta}{dt}$ (rad/s); angular acceleration $\alpha=\dfrac{d\omega}{dt}$ (rad/s$^2$). They are the rotational analogues of $x,v,a$.

    Hint: Analogues of $x,v,a$.

12. 12.Relate linear quantities to angular quantities for a particle at radius $r$.

    $v=r\omega$, tangential acceleration $a_t=r\alpha$, and centripetal acceleration $a_c=\omega^2 r=\dfrac{v^2}{r}$.

    Hint: Multiply angular by $r$.

13. 13.Write the three equations of rotational kinematics for constant $\alpha$.

    $\omega=\omega_0+\alpha t$; $\theta=\omega_0 t+\tfrac{1}{2}\alpha t^2$; $\omega^2=\omega_0^2+2\alpha\theta$. Same form as linear kinematics.

    Hint: Replace $v,a,s$ by $\omega,\alpha,\theta$.

14. 14.Define torque (moment of force) about a point.

    $\vec{\tau}=\vec{r}\times\vec{F}$, magnitude $\tau=rF\sin\theta=F\times(\text{perpendicular distance})$. It measures the turning effect of a force. Unit: N m.

    Hint: $\vec r \times \vec F$.

15. 15.What is the moment arm (lever arm) of a force?

    The perpendicular distance from the axis (or point) to the line of action of the force. $\tau=F\times(\text{moment arm})$.

    Hint: Perpendicular distance to line of action.

16. 16.When is the torque of a force zero?

    When the force passes through the axis ($r=0$ effectively, moment arm $=0$) or when $\vec F$ is parallel/antiparallel to $\vec r$ ($\sin\theta=0$).

    Hint: Line of action through axis.

17. 17.Define the moment of inertia of a body about an axis.

    $I=\sum m_i r_i^2$ (or $\int r^2\,dm$), where $r_i$ is the perpendicular distance of each mass element from the axis. It is the rotational analogue of mass. Unit: kg m$^2$.

    Hint: Rotational analogue of mass.

18. 18.Does moment of inertia depend only on mass?

    No. It depends on the total mass, the distribution of mass, and the chosen axis. The farther the mass from the axis, the larger $I$.

    Hint: Mass, its distribution, and the axis.

19. 19.Define the radius of gyration $k$.

    The distance from the axis at which the whole mass, if concentrated, gives the same moment of inertia: $I=Mk^2$, so $k=\sqrt{I/M}$. Unit: metre.

    Hint: $I=Mk^2$.

20. 20.State the theorem of parallel axes.

    $I=I_{cm}+Md^2$, where $I_{cm}$ is about an axis through the COM and $d$ is the perpendicular distance to a parallel axis. It gives $I$ about any axis parallel to the COM axis.

    Hint: $I_{cm}+Md^2$.

21. 21.State the theorem of perpendicular axes and its restriction.

    For a planar (flat) lamina: $I_z=I_x+I_y$, where $x,y$ lie in the plane and $z$ is perpendicular to it, all three through the same point. Valid only for 2-D bodies.

    Hint: $I_z=I_x+I_y$; lamina only.

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