Important Formulas for System of Particles and Rotational Motion

System of Particles is a group of interrelated particles. A rigid body is defined as a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of a rigid body do not change.

• Rotational motion is the motion of an object around a circular pathway in a fixed orbit either at a fixed point or around its axis.
• The variables of motion in the case of rotational motion are Angular Displacement (θ), Angular Velocity (ω), and Angular Acceleration (α).

Read More: NCERT Solutions For Class 11 Physics System of Particles and Rotational Motion

Key Terms: System of Particles, Rotational Motion, Rigid Body, Velocity, Centre of Mass, Torque, Moment of Inertia, Angular Momentum

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Center of Mass

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The center of mass for a system of particles is defined as that point where the entire mass of the system is imagined to be concentrated. If all the external forces that are acting on the body or a system of bodies were to be applied at the center of mass, the state of rest or motion of the system of bodies shall remain unaffected. In simpler terms, the center of mass of a body or a system is its balancing point.

Centre of Mass

Center of Mass of a Two-Particle System

The centre of mass of a two-particle system always lies on the line that joins the two particles. It is somewhere in between the particles.

When two particles are situated at a distance of x₁ and x₂ from the origin:

 \(X_{cm} = \frac{m_1x_1+m_2x_2}{m_1+m_2}\)

Center of Mass of a System of n Particles

\(X_{cm} = \frac{m_1r_1+m_2r_2+ … m_nr_n}{m_1+m_2+….m_n} = \sum_{ i=1}^n \frac{m_ir_i}{\sum_{i-1}^n m_i} = \frac{1}{m}\sum_{ i=1}^n m_ir_i\)

Expression for Center of Mass

\(X_{cm} = \frac{m_1x_1 + m_2x_2 + ... m_xm_n}{m_1+m_2+ ... m_n} = \frac{1}{\sum^n_{i=1}}=M_iX_i\)

\(Y_{cm} = \frac{m_1x_1 + m_2x_2 + ... m_xm_n}{m_1+m_2+ ... m_n} = \frac{1}{\sum^n_{i=1}}=M_iX_i\)

\(Z_{cm} = \frac{m_1x_1 + m_2x_2 + ... m_xm_n}{m_1+m_2+ ... m_n} = \frac{1}{\sum^n_{i=1}}=M_iX_i\)

Where x_i and z_i is the coordinate of the ith particle

Expression of Center of Mass for a Rigid Body

• X_cm = \(\frac{x dm}{\int dm}\), y_cm = \(\frac{y dm}{\int dm}\) and
• Z_cm = \(\frac{z dm}{\int dm}\), ∫dm = M = Mass of rigid body
• X_cm = \(\frac{1}{M}\) ∫ x dm, y_cm = \(\frac{1}{M}\) ∫ y dm and z_cm = \(\frac{1}{M}\) ∫ z dm
• Velocity of COM v_cm = \(\frac{1}{M}\) (m₁v₁ + m₂v₂ + … m_nv_n)
• Acceleration of COM a_cm = \(\frac{1}{M}\) m₁a₁ + m₂a₂ + … m_na_n)
• Impulse = Change in Momentum: ΔP = P₂ – P₁ = ∫ Fx dt
• If F_ext = 0 = \(\frac{dP}{dt}\) = 0 → dP = 0 → P constant. 

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Rotational Motion of a Rigid Body

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Rotational motion occurs when a rigid body rotates about a fixed axis and every particle of the body moves in a circular path. The perpendicular distance from the axis of rotation to a given point is a radius vector.

In the rotation of a body about a fixed axis angular variables are all the same, but linear variables change

V = rω

A = rα

If a body is executing rotation with constant acceleration, the equations can be written as

ω = ω₀ + ωt

ω= ω₀t + \(\frac{1}{2}\) αt²

ω² - ω_o² = 2αt

Here

• θ : angular displacement its unit is radian
• ω₀: initial angular velocity its unit is rad s⁻¹
• ω : final angular velocity its unit is rad s⁻¹
• α : angular acceleration its unit is rad s⁻²
• t : time

Rotational Motion

Read More: System of Particles & Rotational Motion Important Questions

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Torque

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The turning effect of a force around the axis of rotation is called torque. It measures the force that can cause an object to rotate around an axis. Torque causes the object to achieve angular acceleration. It is a vector quantity and its direction depends on the force’s direction on the axis. Torque is of two types namely static and dynamic. Static Torque does not produce angular acceleration while dynamic torque.

Torque = Force x Perpendicular distance of the line of action of a force from axis of rotation

\(\tau\) = Fd

\(\tau\) = r Fsinθ

or

\(\tau\) = \(\overrightarrow{r}\) x \(\overrightarrow{F}\)

Power of Torque 

Power of a Torque = Torque x Angular Velocity

P = \(\tau\)ω

Work Done by Torque 

Work Done by Torque = Torque x Displacement

W = \(\tau\)θ

Read More: Unit of Torque

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Moment of Inertia

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The inability of a body to change its state of rotation by itself is called the moment of inertia. It is the sum of the products of the masses of various particles and the square of their perpendicular distance from the axis of rotation.

I = m₁r₁² + m₂r₂² + … m_nr_n² or I = \(\sum_{i=1}^n\)m_i x r_i²

Radius of Gyration

Distance from the axis of rotation to a point where the whole mass of the rotating body is supposed to be concentrated. It is denoted as ‘K’. 

If ‘K’ is the radius of gyration, then

 I = mK²

Important Points

• Moment of Inertia depends on the mass of the body and its distribution about the axis of rotation.
• Moment of inertia changes with a change in position of the axis of rotation.
• The Radius of Gyration is not constant. Its value changes with the change in the location of the axis of rotation.

Read More: System of Particles and Rotational Motion MCQs 

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Angular Momentum

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The moment of linear momentum of a particle in rotation about the axis of rotation is known as Angular Momentum.

Angular Momentum = Linear momentum x Perpendicular distance from the axis of rotation

L = r x P

• L = rp sin θ
• R = position vector
• R sin θ: perpendicular distance.
• Relation between L and I: L = Iω

Law of Conservation of Angular Momentum

\(\tau\) = \(\frac{dL}{dt}\)

If \(\tau\) = 0

Where \(\frac{dL}{dt}\) = 0 → L is constant.

L = Iω (Iω is constant)

I₁ω₁ = I₂ω₂

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Rotational Kinetic Energy of an Object

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Rotational Kinetic Energy of an object is dependent on the object’s angular velocity, moment of inertia, and radius and is denoted using the following equation:

KE_rotational = \(\frac{1}{2}\) Iω²

Relation between Angular Momentum and Rotational Kinetic Energy is: 

KE = \(\frac{L^2}{2l}\)

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Solved Examples

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Ques. A disc has a speed of 1800 rpm and it is made to slow down at a uniform rate of 3 rad s^-2. What will be the number of revolutions it makes before coming to rest?

Ans. Given

f=1800 rpm = 30 rotations per sec

• Initial angular velocity of the disc (ω₀) = 2πf = 2π x 30 = 60π rad s^-1
• Final angular velocity of the disc (ω_f) = 0
• Angular acceleration ((α) = -3 rad s^-2

Now

ω_f = ω₀ + αt

or t = ω₀/α = 20π sec

Now

θ = ω₀t +1/2αt²

θ = (60π x 20π) + 1/2(-3)(20π)²

θ = 1200π² – 600π² 

θ =  600π²​ rad

Therefore, Number of revolutions= \(\frac{600 \pi ^2}{2 \pi}\) = 942

Ques. A disc has a speed of 600 rpm and it is made to slow down at a uniform rate of 2 rad s^-2. What will be the number of revolutions it makes before coming to rest?

Ans. Given

f=600 rpm = 10 rotations per sec

• Initial angular velocity of the disc (ω₀) = 2πf = 2π x 10 = 20π rad s^-1
• Final angular velocity of the disc (ω_f) = 0
• Angular acceleration ((α) = -2 rad s^-2

Now

ω_f = ω₀ + αt

or t = ω₀/α = 10π sec

Now

θ = ω₀t +1/2αt²

θ = (20π x 10π) + 1/2(-2)(10π)²

θ = 200π² – 100π² 

θ =  100π²​ rad

Therefore, Number of revolutions= \(\frac{100 \pi ^2}{2 \pi}\) = 157

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Things to Remember

• When a rigid body rotates about a fixed axis, every particle of the body tends to move in a circular path.
• Rotational motion is defined as the motion of a body around a fixed axis.
• The moment of Inertia is the sum of the products of masses of various particles and the square of their perpendicular distance from the axis of rotation.
• Torque is the turning effect about the axis of rotation.
• The movement of Linear Momentum about the axis of rotation is Angular momentum.

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Previous Year Questions

1. If the angular momentum of the system, calculated about O and P are denoted by LoLo and LpLp respectively, then….[JEE Advanced 2012]
2. Assume that (i) they land back on the disc before the disc has completed 1/8 rotation, (ii) their range is less than half the disc radius, and (iii) ωω remains constant throughout. Then….[JEE Advanced 2012]
3. Find its radius of gyration.​..[JKCET 2017]
4. The correct vector relation between linear velocity →v and angular velocity →ω in rigid body dynamics is ( where →r is the position vector)...[JKCET 2009]
5. The diameter of the ring is..[JKCET 2008]
6. The approximate speed of the centre of mass of the pair is...[JKCET 2012]
7. The angular velocity of the platform ω(t) will vary with time t as…..[JEE Advanced 2002]
8. Two objects, each of mass m, are attached gently to the opposite ends of a diameter of the ring. The wheel now rotates with an angular velocity..[JEE Advanced 1983]
9. If the angular acceleration of the wheel is 4πrad/s2, then the moment of Inertia of the wheel is...[JKCET 2011]
10. A cubical block of side a moving with velocity v on a horizontal smooth plane as shown. It hits a ridge at point O. The angular speed of the block after it hits O is...[JEE Advanced 1999]
11. A horizontal force F is applied on the block as shown. If the coefficient of friction is sufficiently high, so that the block does not slide before toppling, the minimum force required to topple the block is..[JEE Advanced 2000]
12. The magnitude of angular momentum of the disc about the origin O is...[JEE Advanced 1999]
13. A solid sphere is in pure rolling motion on an inclined surface having inclination?...[JEE Advanced 2006]
14. Which is perpendicular to plane of the disc), is also equal to I, then the value of r is equal to..[JEE Advanced 2006]
15. The magnitude of its angular momentum with respect to the origin….​[AMUEEE 2018]
16. The spring remains horizontal. If the body is made to rotate at an angular velocity of 2 rad/s, then the elongation of the spring will be...[AMUEEE 2018]
17.  If a quarter part of the plate (shown as shaded) is removed, the centre of mass of the remaining plate would lie at...[AMUEEE 2018]
18. A mass m is moving with a constant velocity along a line parallel to the x-axis, away from the origin. Its angular momentum with respect to the origin...[JEE Advanced 1997]
19. The angle between the force and the momentum is….[JEE Advanced 2007]
20. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at its maximum height h is...[JEE Advanced 1990]