Kinetics of Rigid Bodies: Work and Energy - Theory & Concepts

Kinetics of Rigid Bodies: Work and Energy

Learning Objectives

Understand the principle of work and energy as it applies to rigid bodies.
Calculate the total kinetic energy of a rigid body, including both translational and rotational components.
Determine the work done by forces and couple moments on a rigid body.
Apply power formulas for both forces and moments.
Evaluate the potential energy of a rigid body, incorporating gravity and elastic members.
Solve rigid body motion problems using the conservation of energy principle.

The principle of work and energy for rigid bodies extends the particle concept to include rotational kinetic energy and the work done by couples. This method is particularly useful for problems involving displacement and velocity without needing to solve for acceleration. It provides a scalar approach to relate the forces acting on a body to changes in its speed and elevation.

Civil Engineering Applications

Work-energy methods are applied extensively in structural and mechanical dynamics:

Heavy Machinery & Cranes: Evaluating the motor power required to lift and rotate heavy loads within specific time frames.
Flywheels & Hoists: Sizing flywheels that store rotational kinetic energy to smooth out power fluctuations in industrial lifting mechanisms.
Impact Analysis: Analyzing pendulums and falling masses (like pile drivers) where gravitational potential energy converts directly into large translational and rotational kinetic energies before impact.

Kinetic Energy

Kinetic Energy

The total kinetic energy of a rigid body in general plane motion is the scalar sum of the translational kinetic energy of its center of mass and the rotational kinetic energy about its center of mass.

Kinetic Energy Formula

Calculates the total kinetic energy of a rigid body in general plane motion.

T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2

Variables

Symbol	Description	Unit
$T$	Total kinetic energy	J
$m$	Mass of the rigid body	kg
$v_G$	Velocity of the center of mass	m/s
$I_G$	Mass moment of inertia about the center of mass	kg·m²
$\omega$	Angular velocity of the body	rad/s

Special Cases of Kinetic Energy

Depending on the type of motion, the kinetic energy expression can be simplified:

Translation Only: When the body does not rotate ( $\omega = 0$ ), the rotational term drops out.
Rotation about a Fixed Axis ( $O$ ): The velocity of the center of mass is related to the angular velocity by $v_G = r_G \omega$ . By using the Parallel Axis Theorem ( $I_O = I_G + m r_G^2$ ), the kinetic energy simplifies to $T = \frac{1}{2} I_O \omega^2$ .

Kinetic Energy for Translation

Simplified kinetic energy when the rigid body only translates.

T = \frac{1}{2} m v_G^2

Variables

Symbol	Description	Unit
$T$	Kinetic energy	J
$m$	Mass	kg
$v_G$	Velocity of center of mass	m/s

Kinetic Energy for Fixed Axis Rotation

Simplified kinetic energy when the rigid body rotates about a fixed point O.

T = \frac{1}{2} I_O \omega^2

Variables

Symbol	Description	Unit
$T$	Kinetic energy	J
$I_O$	Mass moment of inertia about the fixed axis O	kg·m²
$\omega$	Angular velocity	rad/s

Work of Forces and Couples

Work

Work is the scalar quantity that represents the energy transferred by forces or couple moments over a displacement or an angular rotation.

Work of a Force

Calculates the work done by a force over a displacement path.

U = \int \mathbf{F} \cdot d\mathbf{r}

Variables

Symbol	Description	Unit
$U$	Work done by the force	J
$\mathbf{F}$	Force vector	N
$d\mathbf{r}$	Differential displacement vector	m

Work of a Couple Moment

Calculates the work done by a couple moment over an angular displacement.

U_M = \int_{\theta_1}^{\theta_2} M \, d\theta

Variables

Symbol	Description	Unit
$U_M$	Work done by the couple moment	J
$M$	Couple moment	N·m
$\theta_1, \theta_2$	Initial and final angular positions	rad

Work of a Constant Couple Moment

Simplified calculation when the couple moment is constant over the rotation.

U_M = M(\theta_2 - \theta_1)

Variables

Symbol	Description	Unit
$U_M$	Work done by the constant couple moment	J
$M$	Constant couple moment	N·m
$\theta_1, \theta_2$	Initial and final angular positions	rad

Power for Rigid Bodies

Power

In rigid body mechanics, power is the rate at which work is done with respect to time. It includes both the power generated by a translating force and the power generated by a rotating couple moment.

Power of a Force

The rate at which a force does work.

P_F = \mathbf{F} \cdot \mathbf{v}_G

Variables

Symbol	Description	Unit
$P_F$	Power of the force	W
$\mathbf{F}$	Force vector	N
$\mathbf{v}_G$	Velocity vector of the center of mass	m/s

Power of a Couple Moment

The rate at which a couple moment does work.

P_M = M \omega

Variables

Symbol	Description	Unit
$P_M$	Power of the couple moment	W
$M$	Couple moment	N·m
$\omega$	Angular velocity	rad/s

Total Mechanical Power

The total mechanical power for a rigid body in general plane motion.

P = \mathbf{F} \cdot \mathbf{v}_G + M \omega

Variables

Symbol	Description	Unit
$P$	Total mechanical power	W
$\mathbf{F}$	Net force	N
$\mathbf{v}_G$	Velocity of the center of mass	m/s
$M$	Net couple moment	N·m
$\omega$	Angular velocity	rad/s

Interactive Simulation

Use the simulation below to explore the rolling motion of a sphere and how kinetic energy involves both translation and rotation.

Rigid Body Incline Rolling (Inertia Race)

Ramp & Incline

Incline Angle (θ)30°

Ramp Length (L)10.0 m

Enable Shape Race Mode

Shape Selection

Dynamics Equations

Acceleration Formula:

a = \frac{g \sin\theta}{1 + \beta}

a₁: 3.50 m/s² (t_end: 2.39s)

Speed at bottom:

v_{max} = \sqrt{\frac{2 g L \sin\theta}{1 + \beta}}

Shape 1 v_max:8.37 m/s

Energy Conservation BalanceTotal: 98.1 J

PE: 100%

Potential Energy of a Rigid Body

Conservative Forces

Forces whose work depends only on the initial and final positions of the body, and not on the path taken. Gravity and ideal springs are common examples.

Potential Energy Types

Similar to particles, when rigid bodies are acted upon by conservative forces, we define potential energy to simplify work-energy calculations. The two primary types are Gravitational Potential Energy ( $V_g$ ) and Elastic Potential Energy ( $V_e$ ).

Gravitational Potential Energy

The potential energy of a rigid body due to its elevation in a gravity field, measured at its center of mass.

V_g = m g y_G

Variables

Symbol	Description	Unit
$V_g$	Gravitational potential energy	J
$m$	Mass of the rigid body	kg
$g$	Acceleration due to gravity	m/s²
$y_G$	Vertical position of the center of mass from a datum	m

Elastic Potential Energy

The potential energy stored in an elastic member, like a spring attached to the rigid body.

V_e = \frac{1}{2} k s^2

Variables

Symbol	Description	Unit
$V_e$	Elastic potential energy	J
$k$	Spring stiffness constant	N/m
$s$	Deformation (stretch or compression) from unstretched length	m

Conservation of Energy for Rigid Bodies

When a rigid body is subjected only to conservative forces (such as weight and spring forces), the total mechanical energy of the body remains constant throughout its motion.

Conservation of Energy Equation

States that the sum of initial kinetic and potential energies equals the final sum.

T_1 + V_1 = T_2 + V_2

Variables

Symbol	Description	Unit
$T_1, T_2$	Initial and final kinetic energies	J
$V_1, V_2$	Initial and final potential energies (gravity + elastic)	J

Friction in Rolling Motion

When a rigid body rolls without slipping, the point of contact is instantaneously at rest ( $v=0$ ). Because work requires displacement, the static friction force does no work. Therefore, a wheel rolling without slipping on an incline still conserves mechanical energy, even though friction is present to prevent slip. However, if the wheel slips, the kinetic friction force does work, and mechanical energy is no longer conserved ( $T_1 + V_1 + U_{NC} = T_2 + V_2$ ).

Key Takeaways

Rotational Kinetic Energy ( $T_{rot} = \frac{1}{2} I \omega^2$ ) must be included in energy calculations for rigid bodies.
Total Kinetic Energy ( $T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2$ ) accounts for both translation of the center of mass and rotation about it.
Work of a Couple ( $U_M = \int M d\theta$ ) adds energy to the system by increasing rotational speed.
Potential Energy of a Rigid Body ( $V_g = mgy_G$ ) is based solely on the elevation of its mass center.
Rolling Without Slip relates linear and angular velocity ( $v = r\omega$ ), simplifying energy expressions without static friction doing work.
Conservation of Energy applies when conservative forces (gravity, springs) are the primary workers.

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