Physics
9
Centre of Mass - Definition
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The center of mass of a system of particles is a special point that moves as if all the mass of the system were concentrated at that point and all external forces were applied there.
['X = (1/M) * Σ(m_i * x_i)', 'Y = (1/M) * Σ(m_i * y_i)', 'Z = (1/M) * Σ(m_i * z_i)', 'M = Σm_i', 'R_CM = (1/M) * Σ(m_i * r_i)']
Let us consider a collection of N particles. Let the mass of the ith particle be m_i and its coordinates with reference to the chosen axes be x_i, y_i, z_i. The coordinates of the center of mass (X, Y, Z) are given by:
X = (1/M) * Σ(m_i * x_i)
Y = (1/M) * Σ(m_i * y_i)
Z = (1/M) * Σ(m_i * z_i)
Where M = Σm_i is the total mass of the system.
The position vector of the center of mass R_CM is given by:
R_CM = (1/M) * Σ(m_i * r_i)
Elastic Collision Approximations
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Approximations for elastic collisions when one mass is much larger than the other.
v1' ≈ v1, v2' ≈ 2v1 - v2
When m1 >> m2, v1' ≈ v1 and v2' ≈ 2v1 - v2. The heavier body continues with almost the same velocity, and the velocity of the lighter body changes significantly. If the heavier body is at rest (v1 = 0), the lighter body returns with almost the same speed (v2' = -v2).
Elastic Collision of Equal Masses
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In an elastic collision between two bodies of equal mass, their velocities are interchanged.
v1' = v2, v2' = v1
When two bodies of equal mass collide elastically, their velocities are mutually interchanged.
Perfectly Inelastic Collision in One Dimension
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When objects stick together after a collision.
V = (m1v1 + m2v2) / (m1 + m2), Loss in KE = 1/2 * m1v1^2 + 1/2 * m2v2^2 - 1/2 * (m1 + m2) * V^2
When perfectly inelastic bodies moving along the same line collide, they stick to each other. The final common velocity and the loss in kinetic energy are calculated based on the conservation of linear momentum.
Coefficient of Restitution
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A measure of how elastic a collision is.
e = (velocity of separation) / (velocity of approach)
The coefficient of restitution (e) is defined as the ratio of the velocity of separation to the velocity of approach. It ranges from 0 to 1, where e = 1 represents a perfectly elastic collision and e = 0 represents a perfectly inelastic collision.
Elastic Collision in Two Dimensions
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Description of elastic collisions in two dimensions.
None
Considers two objects with masses m1 and m2. Initially, m2 is at rest. Object m1 moves along the x-axis and collides with m2.