Motion of System of Particles and Rigid Body

The One Thing

The motion of a system can be separated into translation of its centre of mass and rotation about an axis. Translational motion is governed by external force and linear momentum, while rotational motion is governed by torque, angular momentum, and moment of inertia.

Reactions, Processes and Experiments

No named reactions, experiments, or experimental processes are included in the summary data.

Key Terms

• System of particles: A collection of particles considered together for studying their combined motion.
• Centre of mass: The point at which the entire mass of a system may be considered concentrated for describing translational motion.
• Centre of mass position: For particles with masses $m_i$ and position vectors $r_i$, $R=(\Sigma m_i{r}_i)/(\Sigma m_i)$. For two particles, $R=(m_1{r}_1+m_2{r}_2)/(m_1+m_2)$.
• Linear momentum: The product of mass and velocity; for a particle, $p=mv$, and for a system, total momentum is $P=\Sigma m_i{v}_i$.
• Momentum of a system: The total momentum of a system equals its total mass multiplied by the velocity of its centre of mass: $P=M{V}_{CM}$.
• External force: A force acting on a system from outside it; it changes the total momentum of the system.
• Conservation of linear momentum: If the net external force on a system is zero, its total linear momentum remains constant.
• Rigid body: An ideal body in which the distance between every pair of particles remains constant during motion.
• Angular displacement: The angle through which a body rotates, usually represented by theta and measured in radians. In radians, theta = arc length/radius.
• Angular velocity: The rate of change of angular displacement: omega = dtheta/dt.
• Angular acceleration: The rate of change of angular velocity: alpha = domega/dt or alpha = d^2theta/dt^2.
• Torque: The turning effect of a force about an axis or point, given by tau = r x F; its magnitude is tau = rF sin theta.
• Moment arm: The perpendicular distance from the axis of rotation to the line of action of the force.
• Couple: A pair of equal and opposite parallel forces acting along different lines, producing rotation without translation.
• Moment of inertia: The rotational analogue of mass, measuring how strongly a body resists angular acceleration: $I=\Sigma m_i{r}_i^2$ for discrete particles and $I=\int r^{2}dm$ for a continuous body.
• Radius of gyration: The distance $k$ from the axis at which the whole mass may be assumed concentrated to give the same moment of inertia: $I=M{k}^2$.
• Angular momentum: The rotational analogue of linear momentum; for a particle, $L=rxp$, and for a rigid body rotating about a fixed axis, $L=Iomega$.
• Rotational form of Newton's second law: The net external torque equals the rate of change of angular momentum; for a fixed axis, tau = I alpha.
• Rotational kinetic energy: The energy possessed due to rotation: $K_{rot}=1/2Iomeg{a}^2$.
• Rolling motion: A combination of translation and rotation in which a body rolls without slipping when $v_{CM}=Romega$.
• Parallel-axis theorem: The moment of inertia about any parallel axis is $I=I_{CM}+M{d}^2$, where $d$ is the distance between the axes.
• Perpendicular-axis theorem: For a plane lamina, the moment of inertia about an axis perpendicular to its plane is the sum of the moments of inertia about two mutually perpendicular axes in the plane: $I_z=I_x+I_y$.
• Equilibrium of a rigid body: A rigid body is in equilibrium when both the net external force and the net external torque are zero.
• Impulse: The change in momentum: $J=\int Fdt=Deltap$; for constant force, $J=FDeltat$.
• Moment of inertia of a thin ring: About its central axis, $I=M{R}^2$.
• Moment of inertia of a solid disc or cylinder: About its symmetry axis, $I=1/2M{R}^2$.
• Moment of inertia of a solid sphere: About a diameter, $I=2/5M{R}^2$.
• Moment of inertia of a thin spherical shell: About a diameter, $I=2/3M{R}^2$.
• Moment of inertia of a thin rod: About its centre, $I=1/12M{L}^2$; about one end, $I=1/3M{L}^2$.
• Centre of gravity: In a uniform gravitational field, the centre of gravity coincides with the centre of mass.
• Right-hand rule: The direction of angular velocity, angular acceleration, torque, and angular momentum is determined using the right-hand rule.

Easily Confused

• Centre of mass and centre of gravity: They coincide in a uniform gravitational field; the centre of mass describes mass distribution, whereas the centre of gravity describes the effective point of action of gravitational force.
• Torque and work: Both have the same dimensions and SI unit, $Nm$, but torque is a vector turning effect whereas work is a scalar energy transfer.
• Mass and moment of inertia: Mass resists linear acceleration, whereas moment of inertia resists angular acceleration and depends on both mass and its distance from the axis.
• Angular momentum and linear momentum: Linear momentum is $p=mv$, whereas angular momentum is $L=rxp$ or $L=Iomega$ for a rigid body about a fixed axis.
• Translational and rotational kinetic energy: Translational kinetic energy is associated with motion of the centre of mass, whereas rotational kinetic energy is $K_{rot}=1/2Iomeg{a}^2$.
• Equilibrium of force and equilibrium of a rigid body: Zero net force prevents translational acceleration, but complete rigid-body equilibrium also requires zero net torque.
• Rolling motion and pure translation: Rolling motion combines translation and rotation; for rolling without slipping, $v_{CM}=Romega$.

What Gets Asked

• Centre-of-mass calculations: Questions may require use of $R=(\Sigma m_i{r}_i)/(\Sigma m_i)$ or $R=(m_1{r}_1+m_2{r}_2)/(m_1+m_2)$. Marks are lost by using an inappropriate total mass or position vector.
• Motion of a system under external forces: Questions may use $F_{ext}=dP/dt=M{a}_{CM}$. The key error is treating internal forces as contributing to the net force of the complete system; they occur in equal and opposite pairs and cancel.
• Momentum and impulse: Problems may require $P=M{V}_{CM}$, conservation of linear momentum when the net external force is zero, or $J=\int Fdt=Deltap$. Marks are lost by applying momentum conservation when a non-zero external force acts.
• Rotational kinematics: Questions may require $omega=omeg{a}_0+alphat$, $theta=omeg{a}_{0}t+1/2alpha{t}^2$, or $omeg{a}^2=omeg{a}_0^2+2alphatheta$. These equations apply for constant angular acceleration.
• Torque and rotational dynamics: Questions may require $tau=rFsintheta$, $tau=Ialpha$, or $tau=dL/dt$. The common error is using the full distance $r$ instead of the perpendicular moment arm, or omitting the angle in $rFsintheta$.
• Rolling and rotational energy: Questions may require $K=1/2M{v}_{CM}^2+1/2I_{CM}omeg{a}^2$ together with $v_{CM}=Romega$. Marks are lost by including only translational or only rotational kinetic energy.
• Rigid-body equilibrium and moment of inertia: Questions may require $\Sigma F=0$, $\Sigma tau=0$, the parallel-axis theorem $I=I_{CM}+M{d}^2$, or the perpendicular-axis theorem $I_z=I_x+I_y$. The central distinction is that both force and torque conditions are required for complete equilibrium.