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

Kinetics of Rigid Bodies: Work and Energy

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.

Kinetic Energy

The total kinetic energy of a rigid body in general plane motion is the sum of translational and rotational kinetic energy:

Kinetic Energy Formula

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

Where:

$v_G$ is the velocity of the center of mass.
$I_G$ is the mass moment of inertia about the center of mass.
$\omega$ is the angular velocity.

Special Cases

Translation Only: $\omega = 0 \Rightarrow T = \frac{1}{2} m v_G^2$
Rotation about Fixed Axis $O$ : The velocity of the center of mass is $v_G = r_G \omega$ .
$T = \frac{1}{2} m (r_G \omega)^2 + \frac{1}{2} I_G \omega^2 = \frac{1}{2} (I_G + m r_G^2) \omega^2 = \frac{1}{2} I_O \omega^2$
(Using Parallel Axis Theorem: $I_O = I_G + m r_G^2$ ).

Work of Forces and Couples

Work Formulas

Work of a Force: Same as particles, $U = \int \mathbf{F} \cdot d\mathbf{r}$ .
Work of a Couple Moment ( $M$ ): $U_M = \int_{\theta_1}^{\theta_2} M \, d\theta$ If $M$ is constant: $U_M = M(\theta_2 - \theta_1)$ .

Power for Rigid Bodies

In rigid body mechanics, power is the rate at which work is done. It includes the power generated by a force (as seen in particles) and the power generated by a couple moment.

Power Formula

The power generated by a force

\mathbf{F}

is:

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

The power generated by a couple moment

M

is:

P_M = M \omega

The total mechanical power for a rigid body in general plane motion is the sum of both:

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

Interact with the simulation below to explore the rolling motion of a sphere.

Rolling Sphere Simulator

Height (

h

)2.0 m

Incline Angle (

\theta

)30°

Current Time (

t

)0.00 s

Current Velocity (

v

)0.00 m/s

Max Velocity (v_max)5.29 m/s

Potential Energy of a Rigid Body

Similar to particles, when rigid bodies are acted upon by conservative forces, we can define potential energy to simplify the work-energy calculations.

Types of Potential Energy

Gravitational Potential Energy ( $V_g$ ): The potential energy of a rigid body due to gravity is determined by the vertical position of its center of mass $G$ . $V_g = m g y_G$ Where $y_G$ is the vertical coordinate of the mass center measured from an arbitrary datum.
Elastic Potential Energy ( $V_e$ ): The potential energy stored in an elastic member, like a spring, attached to the rigid body. It depends only on the deformation $s$ of the spring from its unstretched length. $V_e = \frac{1}{2} k s^2$

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.

Conservation Equation

T_1 + V_1 = T_2 + V_2

Where

V = V_g + V_e

This means the initial kinetic plus initial potential energy equals the final kinetic plus final potential energy.

Important

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 and rotation.
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.
Conservation of Energy applies when conservative forces (gravity, springs) are the primary workers. Friction in pure rolling does no work.

PreviousKinetics of Rigid Bodies: Force and Acceleration - Examples & Applications

Quiz Me

NextKinetics of Rigid Bodies: Work and Energy - Examples & Applications