System of Particles and Rotational Motion

A system of particles is a collection of different particles interacting with one another.

• The theory of the system of particles is used to calculate the force and center of mass of rigid bodies using pulleys, multiple body blocks, and other parameters. 
• It is essential in classical mechanics and is easily connected to Newtonian physics.
• Rotational motion is the combination of circular motion of a large number of particles of a rigid body.
• A rigid body is one in which the spacing between any constituent particles stays constant when an external force is applied.
• The center of mass is a point where the distributed mass of an object is zero or a point where an object balances itself.
• System of particles and rotational motion are very important concepts to understand to overall motion of a rotating body.

Key Terms: Rotational motion, Angular momentum, Force, Moment of inertia, Kinetic energy, Rigid body, Rolling motion, Center of mass, Equation of motion, Center of gravity

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System of Particles

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A collection of a large number of particles interacting with each other is known as a system of particles.

• Every particle of a system of particles exerts a force on each other particles, known as internal force.
• These forces cancel out in pairs and there is no effect on the overall motion of the particles.

Types of System of Particles

There are two types of systems of particles

1. Discrete system: In the discrete system, all bodies are separated from one another so that there is no point of contact. An auditorium with people sitting in a random pattern is a good example of a discrete system.
2. Continuous system: All bodies in the continuous system are arranged compactly with minimal space between them. In this case, the center of motion plays an essential role in determining the type of work done and the point of force application.

System of particles

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Center of Mass

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“The center of mass of a body is the point where the whole mass of the system were supposed to be concentrated and this point has translational motion as the system as a whole”.

• The concept of the center of mass of a system describes the overall motion of the system.
• It replaces the system with an equivalent point, where the entire mass of the body is supposed to be concentrated.

The formula of the center of mass of a system of n particles of masses m₁, m₂, m₃,......., m_n respectively along a straight line at distances x₁, x₂, x₃,......., x_n from the origin is given by

\(X_{CM} = \frac{m_1x_1+m_2x_2+..........+m_nx_n}{m_1+m_2+........+m_n}\)

Center of mass

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Equation of Motion of Center of Mass

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Consider a system of n particles of masses m₁, m₂, m₃,.......m_n. Let r₁, r₂, r₃,.......r_n be the respective positions, then the position of the center of mass is given by

\(R_{CM} = \frac{m_1x_1+m_2x_2+..........+m_nx_n}{m_1+m_2+........+m_n}\)

⇒ R_CM(m₁ + m₂ +.....+ m_n) = m₁r₁ + m₂r₂ +.....+ m_nr_n

But (m₁ + m₂ +.....+ m_n) = M, the mass of the system

⇒ R_CMM = m₁r₁ + m₂r₂ +.....+ m_nr_n

Differentiating both sides with respect to t, we get

V_CMM = m₁v₁ + m₂v₂ +.....+ m_nv_n

Where

• V_CM is the velocity of the center of mass
• v₁, v₂,......v_n are the velocities of the n particles

Again differentiating both sides with respect to t, we get

a_CMM = m₁a₁ + m₂a₂ +.....+ m_na_n

Where

• a_CM is the acceleration of the center of mass
• a₁, a₂,......a_n are the accelerations of the n particles

⇒ a_CMM = F₁ + F₂ +........+ F_n

Where F₁, F₂,......., F_n are the external forces

The center of mass of the system of particles moves as if whole the mass of the system were concentrated at the position of the center of mass and all the external forces were applied at the center of mass.

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Rigid Body

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A rigid body is one in which the distance between any two constituent particles remains fixed under the application of external force.

A rigid body has two types of motion

• Translational Motion: A rigid body has translational motion if it moves on a horizontal surface in such a way that every particle of the rigid body has the same velocity.
• Rotational Motion: A rigid body has rotational motion if it rotates about a fixed axis such that every particle of the body moves in a circle with its center of axis of rotation.

A rigid body moving over a horizontal surface has translational and rotational motion simultaneously.

Translational and Rotational Motion

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Vector Product of Vectors

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The vector product or cross product of two vectors is a single vector whose magnitude is equal to the product of the magnitudes of two given vectors multiplied by the sine of the smaller angle between them.

\(\vec{C} = \vec{A} \times \vec{B} = AB sin \theta\) 

Properties of Vector Product

The following are the properties of vector product

• It does not obey commutative law

i.e. \(\vec{A} \times \vec{B}\) ≠ \(\vec{B} \times \vec{A}\)

But \(\vec{A} \times \vec{B}\) = – \(\vec{B} \times \vec{A}\)

• It obeys the distributive law

i.e. \(\vec{A} (\vec{B} + \vec{C})\) = \(\vec{A} \times \vec{B}\) + \(\vec{A} \times \vec{C}\)

• The cross product of two parallel or equal vectors is zero.

i.e. \(\vec{A} \times \vec{B}\) = AB sin0 = 0 or \(\vec{A} \times \vec{A}\) = AA sin0 = 0

• The cross product of two perpendicular vectors is equal to the product of the magnitude of the two vectors.

i.e. \(\vec{A} \times \vec{B}\)=\(\vec{A} \times \vec{B}\) sin90 = AB

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Equilibrium of Rigid Bodies

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A rigid body is said to be in equilibrium if the net external force or torque acting on it does not change the translational or rotational state of the body.

This means the body has neither linear acceleration nor angular acceleration.

There are two conditions for the equilibrium of rigid bodies: 

• Condition for Translational Equilibrium
• Condition for Rotational Equilibrium

Condition for Translational Equilibrium

A rigid body is said to be in translational equilibrium 

• If the body remains at rest or moving with constant velocity.
• If the net external force acting on the body is zero.

∏F_ex = 0

• If its total linear momentum remains constant i.e. does not change with time.

\(\vec{p}\) = constant

Condition for Rotational Equilibrium

A rigid body is said to be in rotational equilibrium 

• If the body does not rotate or rotate with uniform angular velocity.
• If the net external torque acting on the body is zero.

∏∏_ex = 0

• If its total angular momentum remains constant i.e. does not change with time.

\(\vec{L}\) = constant

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Rotational Motion

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Rotational motion is defined as the motion of an object in a fixed orbit around a circular path.

• If any line drawn on the rigid body does not remain parallel to itself throughout its motion, then the body is said to be in rotational motion.
• Examples of rotational motion are ceiling fans, bicycle wheels, the rolling of football on the ground, etc.
• There should be at least one point on the body that remains fixed in rotational motion.
• An imaginary line drawn perpendicular to the plane of motion of different points of the body and passing through the stationary point is called the axis of rotation.

Rotational Variables

To describe the rotational motion of a rigid body, certain parameters such as position, velocity, and acceleration are needed.

• Angle of rotation (θ): An angle of rotation is the amount a figure is rotated about a fixed point known as a point of rotation.
• Angular velocity (ω): The angular velocity of an object is the rate at which it rotates or revolves about an axis.
• Angular acceleration (ɑ): Angular acceleration is the time rate of change of angular velocity.
• Angular momentum (L): The property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object is known as angular momentum.

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Relation Between Linear and Angular Variables

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Consider a body rotating about a fixed in a circular orbit of radius r.

The relation between linear and angular velocities is given by

v = rω

Where

• v is the linear velocity
• ω is the angular velocity

The relation between linear and angular accelerations is given by

a = rα

Where

• a is the linear acceleration
• α is the angular acceleration

The relation between linear and angular momentum is given by

L = rp

Where

• p is the linear momentum
• L is the angular momentum

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Moment of Inertia

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The moment of inertia of a body about a given axis of rotation is defined as the sum of the product of the masses of the constituting particles and squares of their distances from the axis of rotation.

• It is also known as Rotational Inertia.
• The SI unit of moment of inertia is kg m²
• The moment of inertia plays the same role in rotational motion as mass does in linear motion.

The formula of the moment of inertia is given by

I = m₁r₁² + m₂r₂² + m₃r₃² + .......… + m_nr_n²

Where

• I is the moment of inertia of the body
• m₁, m₂,...., and m_n be the mass of the constituent particles.
• r₁, r₂,...., and r_n are the distances of the constituent particles from the axis of rotation.

Relation Between Angular Momentum and Moment of Inertia

The relation between angular momentum and moment of inertia is given by

L = Iω

Where

• L is angular momentum
• I is the moment of inertia
• ω is the angular velocity

Relation Between Torque and Moment of Inertia

The relation between torque and moment of inertia is given by

τ = Iɑ

Where

• τ is torque
• I is the moment of inertia
• ɑ is the angular acceleration

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Theorems on Moment of Inertia

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There are two basic theorems on the moment of inertia. These are

• Theorem of perpendicular axes
• Theorem of parallel axes

Theorem of Perpendicular Axes

According to this theorem, the moment of inertia of a plane lamina about an axis perpendicular to the plane of the lamina is the sum of moments of inertia about any two mutually perpendicular axes both lying in the same plane.

I_z = I_x + I_y

Where

• I_z is the moment of inertia about the z-axis
• I_x is the moment of inertia about the x-axis
• I_y is the moment of inertia about the y-axis

Theorem of Parallel Axes

According to this theorem, the moment of inertia of a plane lamina about any axis in its plane is equal to its moment of inertia about a parallel axis passing through the center of mass of the lamina plus the product of the mass of the lamina and square of the distance between the axes.

I = I_C + Mh²

Where

• I is the moment of inertia about any axis
• I_C is the moment of inertia about a parallel axis passing through the center
• M is the mass of the lamina
• h is the distance between the axes.

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Rolling Motion

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Rolling is a type of motion that combines rotation and translation of an object with respect to a surface, such that the two are in contact with one other without sliding if ideal conditions exist.

• The rolling motion is a combination of translational motion and rotational motion.
• Pure rolling is defined as rolling without sliding.

Rolling motion

The total energy of a rolling body is the sum of its translational kinetic energy and rotational kinetic energy i.e.

E = 1/2 mv² + 1/2 I (v²/r²)

Where

• m is the mass of the body
• v is the velocity of the body
• r is the radius

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Things to Remember

• System of particles is a collection of different particles interacting with each other.
• Rotational motion is the combination of circular motion of a large number of particles of a rigid body.
• Center of mass is a point where the whole mass of the body is supposed to be concentrated.
• The center of weight is a point where the whole weight of the body is concentrated.
• A rigid body is one in which the distance between any two constituent particles remains fixed.
• The property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object is known as angular momentum.

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