16.7
Kinetic Energy for a Rigid Body
Consider a rigid body undergoing a general planar motion. Its center of mass is located at point G.
The kinetic energy of the i-th particle of the rigid body, relative to an arbitrary point A, can be expressed using the relative velocity definition. The position vector rA extends from point A to mass element i.
Using the scalar product, expressing the equation in an integral form, and using a vector identity, the kinetic energy for the entire body is expressed in terms of its angular momentum.
Furthermore, if point A is the center of mass of the rigid body, then the integral of the position vector and mass element becomes zero.
Here, the kinetic energy is expressed as the sum of the kinetic energy of the center of mass of the body and the rotational kinetic energy of the body.
If point A is a fixed point on the rigid body, the kinetic energy equation gets simplified. Using the definition of angular momentum of the rigid body, the kinetic energy equation can be expressed in component form.
Imagine a solid object involved in a general planar movement, with its center of mass pinpointed at a spot labeled G. The object's kinetic energy relative to an arbitrary point A can be quantified for each of its particles - the ith particle in this case. This measurement is achieved through the employment of the relative velocity definition. The position vector, known as rA, extends from point A to the mass element i.
The calculation of the kinetic energy for the entire object involves several steps. First, the scalar product comes into use. Following that, the equation is expressed in its integral form. Lastly, a vector identity is utilized to complete the calculation. The complexity of the kinetic energy equation can be reduced if point A is deemed as a fixed point on the solid object. By applying the definition of the object's angular momentum, the equation can then be represented as follows
An interesting scenario unfolds when point A coincides with the center of mass of the solid object. In this case, the integral of the position vector and the mass element equates to zero. This leads to a simplified expression of the kinetic energy. It is represented as the sum of two components: the kinetic energy of the center of mass of the object and the rotational kinetic energy of the object.
Consider a rigid body undergoing a general planar motion. Its center of mass is located at point G.
The kinetic energy of the i-th particle of the rigid body, relative to an arbitrary point A, can be expressed using the relative velocity definition. The position vector rA extends from point A to mass element i.
Using the scalar product, expressing the equation in an integral form, and using a vector identity, the kinetic energy for the entire body is expressed in terms of its angular momentum.
Furthermore, if point A is the center of mass of the rigid body, then the integral of the position vector and mass element becomes zero.
Here, the kinetic energy is expressed as the sum of the kinetic energy of the center of mass of the body and the rotational kinetic energy of the body.
If point A is a fixed point on the rigid body, the kinetic energy equation gets simplified. Using the definition of angular momentum of the rigid body, the kinetic energy equation can be expressed in component form.
- Kinetic Energy - Energy of an object due to its motion.
- Center of Mass - A position defined relative to an object or system of objects.
- Position Vector - A vector drawn from the origin of a coordinate system to an object's position.
- Rigid Body - An idealized physical object that maintains its shape and size throughout its motion.
- Relative Kinetic Energy - The measurement of kinetic energy related to a specific position.
- Define Kinetic Energy – Understand how it changes with motion (e.g., kinetic energy).
- Contrast Fixed Point vs Center of Mass – Grasp differences between these positions (e.g., center of mass).
- Explore Equations – Comprehend their role in calculating kinetic energy (e.g., integral form).
- Explain Role of Position Vector – Understand its relation to relative kinetic energy.
- Apply Understanding on Rotational Kinetics – Master its impact on a rigid body's kinetic energy.
- What is Kinetic Energy and how does it vary with motion?
- What is the significance of the Position Vector in determining Relative Kinetic Energy?
- What are the differences between a fixed point and a center of mass?
From Chapter 16:
Now Playing
16.7 : Kinetic Energy for a Rigid Body
3-Dimensional Kinetics of a Rigid Body
688 Views
16.1 : Moments and Product of Inertia
3-Dimensional Kinetics of a Rigid Body
1.4K Views
16.2 : Inertia Tensor
3-Dimensional Kinetics of a Rigid Body
1.4K Views
16.3 : Moment of Inertia about an Arbitrary Axis
3-Dimensional Kinetics of a Rigid Body
788 Views
16.4 : Angular Momentum about an Arbitrary Axis
3-Dimensional Kinetics of a Rigid Body
542 Views
16.5 : Angular Momentum and Principle Axes of Inertia
3-Dimensional Kinetics of a Rigid Body
692 Views
16.6 : Principle of Impulse and Moment
3-Dimensional Kinetics of a Rigid Body
713 Views
16.8 : Equation of Motion for a Rigid Body
3-Dimensional Kinetics of a Rigid Body
820 Views
16.9 : Euler Equations of Motion
3-Dimensional Kinetics of a Rigid Body
784 Views
16.10 : Torque Free Motion
3-Dimensional Kinetics of a Rigid Body
978 Views