Questions and Answers
What is the centre of mass?
The centre of mass is the point that behaves as if the whole mass of the system is concentrated at it and all external forces are acting on it.
What happens to the centre of mass of an isolated system?
The centre of mass of an isolated system has a constant velocity.
In symmetrical bodies with homogeneous mass distribution, where does the centre of mass coincide?
What is the moment of inertia denoted by?
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The moment of inertia of a body only depends on its mass.
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The radius of gyration is denoted by ____.
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Match the following concepts with their descriptions:
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State the formula for the moment of inertia about an arbitrary axis according to the Parallel Axes Theorem.
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What condition defines a rigid body?
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Study Notes
Centre of Mass
- The centre of mass (CM) is the point where the total mass of a system can be considered concentrated.
- For rigid bodies, the CM location remains unchanged regardless of the body's motion (rest or accelerated).
- For a system of n particles with masses m1, m2, …, mn, the position vector of the CM can be calculated based on their individual position vectors.
Two Particle System
- In a two-particle system with masses m1 and m2, the position of the CM from mass m2 is given by the formula: (m1 * d) / (m1 + m2).
- The velocity of the CM can be derived from the individual velocities v1 and v2 of the particles.
- The acceleration of the CM can be found using the accelerations a1 and a2 of the particles.
- An isolated system's CM maintains a constant velocity, implying it remains at rest or continues moving uniformly if initially in motion.
Characteristics of Centre of Mass
- CM location is influenced by the shape, size, and mass distribution of the object.
- The CM does not necessarily lie within the physical boundaries of the object.
- For symmetrical bodies with uniform mass distribution, the CM corresponds with the geometric center.
Motion Types
- Translational Motion: Occurs when all particles of a rigid body displace uniformly in the same direction.
- Rotational Motion: Involves particles moving circularly around an axis, maintaining a straight line through the centers of the circles—this is called the axis of rotation.
Rigid Body and Motion
- A rigid body retains its shape; the relative distances between its constituent particles remain constant under force application.
- Rigid body motion involves both translational and rotational components.
Moment of Inertia
- Denoted by I, the moment of inertia represents an object's resistance to changes in its rotational motion.
- It is defined as the sum of the products of masses of particles and the square of their distances from the rotation axis.
- Units: kg·m²; Dimensional formula: [ML²].
- Moment of inertia is affected by:
- Position and orientation of the rotation axis
- Shape and size of the body
- Mass distribution about the axis
Radius of Gyration
- The radius of gyration (K) is the root mean square distance of particles from the axis of rotation.
- The relationship between moment of inertia and radius of gyration is outlined as I = M * K².
Theorems Related to Moment of Inertia
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Parallel Axes Theorem: Moment of inertia about any arbitrary axis equals the moment of inertia about a parallel axis through the CM plus the product of mass and the squared distance between the axes.
- Formula: I = I_CM + M * r²
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Perpendicular Axes Theorem: The moment of inertia of a two-dimensional body about an axis perpendicular to its plane is the sum of the moments of inertia about two mutually perpendicular axes in its plane.
- Formula: I_z = I_x + I_y
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Description
Explore the concept of the centre of mass in a system of particles and rotational motion. This quiz covers important principles, including how the centre of mass behaves under various conditions. Test your understanding of this fundamental physics topic as you prepare for your exams.
