Physics
Centre of Mass
Centre of Mass of Two Particles
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The center of mass of two particles lies on the line joining them, dividing the line in the inverse ratio of their masses.
m1(OC) = m2(CP)
Consider two particles of masses m1 and m2 separated by a distance d. The center of mass lies on the line joining the two particles. If O, C, and P are the positions of m1, the center of mass, and m2 respectively, then m1(OC) = m2(CP). The center of mass divides the line internally in the inverse ratio of their masses.
Centre of Mass of Several Groups of Particles
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The center of mass of a system of multiple groups of particles can be found by treating each group as a point particle located at its center of mass, with mass equal to the total mass of that group.
X = (M1X1 + M2X2) / (M1 + M2), Y = (M1Y1 + M2Y2) / (M1 + M2), Z = (M1Z1 + M2Z2) / (M1 + M2)
Consider a system with two groups of particles, N1 and N2, with total masses M1 and M2 respectively. Let the centers of mass of the two groups be at C1 and C2. The x-coordinate of the center of mass of the combined system is given by: X = (M1X1 + M2X2) / (M1 + M2), where X1 and X2 are the x-coordinates of C1 and C2 respectively. Similarly, Y = (M1Y1 + M2Y2) / (M1 + M2) and Z = (M1Z1 + M2Z2) / (M1 + M2). The overall center of mass is the same as if two point particles of masses M1 and M2 were placed at C1 and C2, respectively.
Centre of Mass of a Uniform Semicircular Wire
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To find the center of mass of a semicircular wire, we consider a small element of length R dq and integrate over the semicircle.
['dm = (M/π) dq', 'X = (1/M) ∫ x dm', 'Y = (1/M) ∫ y dm']
The mass per unit length of the wire is M/πR. The mass of the element is dm = (M/π)dq. The coordinates of the center of mass are:
X = (1/M) ∫ x dm = (1/M) ∫ (R cos θ) (M/π) dθ = 0
Y = (1/M) ∫ y dm = (1/M) ∫ (R sin θ) (M/π) dθ = 2R/π. The center of mass is at (0, 2R/π).
Centre of Mass of a Uniform Semicircular Plate
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To find the center of mass of a semicircular plate, we consider a semicircular wire element of radius r and integrate over the radius from 0 to R.
['dm = (2M/R²) r dr', 'Y = (1/M) ∫ y dm']
The area of the semicircular element is πr dr. The area of the plate is πR²/2. The mass per unit area is M/(πR²/2) = 2M/πR². The mass of the semicircular element is dm = (2M/R²) r dr. The y-coordinate of the center of mass of this wire is 2r/π. The y-coordinate of the center of mass of the plate is:
Y = (1/M) ∫ (2r/π) (2M/R²) r dr = (4/(3π))R. The x-coordinate of the center of mass is zero by symmetry. The center of mass is at (0, 4R/(3π)).
Motion of the Centre of Mass
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The center of mass of a system of particles moves as if all the mass of the system were concentrated at that point and all external forces were applied at that point.
['X = (m1x1 + m2x2) / (m1 + m2)']
Consider two particles A and B of masses m1 and m2. If no external force acts on the two-particle system, the center of mass at time t is situated at X = (m1x1 + m2x2) / (m1 + m2).
Centre of Mass of a System of Particles
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The centre of mass of a system is the point where the entire mass of the system can be assumed to be concentrated. Its location depends on the masses and positions of the individual particles in the system.
Xcm = Σ(mi * xi) / Σmi
Ycm = Σ(mi * yi) / Σmi
The coordinates of the centre of mass (X, Y) for a system of particles are calculated as follows:
X = (m1x1 + m2x2 + m3x3 + ...) / (m1 + m2 + m3 + ...)
Y = (m1y1 + m2y2 + m3y3 + ...) / (m1 + m2 + m3 + ...)
Centre of Mass of a Continuous Object
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For continuous objects, we replace the summation with integration to find the centre of mass. We consider infinitesimal mass elements (dm) and integrate over the entire object.
Xcm = ∫(x dm) / ∫dm
Ycm = ∫(y dm) / ∫dm
The coordinates of the centre of mass (X, Y) for a continuous object are calculated as follows:
X = ∫(x dm) / ∫dm
Y = ∫(y dm) / ∫dm
Centre of Mass Concepts
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The center of mass (COM) of a system of particles is the point that moves as though all of the system's mass were concentrated there and all external forces were applied there. Internal forces do not affect the motion of the COM. In collisions, momentum is always conserved, but kinetic energy is only conserved in elastic collisions.
None explicitly stated in the extract.
- The centre of mass of a composite system is the weighted average of the positions of its constituent parts.
- Internal forces cannot change the linear momentum of a system.
- In collisions, momentum is always conserved.
- Kinetic energy is conserved in elastic collisions, but not in inelastic collisions.
- The motion of the center of mass is independent of internal forces.