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Finding the Center of Mass: Discrete Masses and Explosions

This article explores the concept of center of mass for discrete objects and its application in explosions and thrust calculations. It also discusses the conservation of momentum and kinetic energy before and after separations.

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Finding the Center of Mass: Discrete Masses and Explosions

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  1. Finding the Center

  2. Discrete Center of Mass • Discrete masses are treated as separate objects. • They can lie on a line or in a plane. m2 m3 rCM m1 m4

  3. Discrete Problem • Two trucks are on a barge. Find the center of mass. • m1 = 11 Mg, d1 = -12 m • m2 = 23 Mg, d2 = +12 m • mb = 35 Mg • m1d1 = -132 Mg m • m2d2 = +276 Mg m • mbdb = 0 • M = 69 Mg • dCM = 2.1 m m1 m2 mb d1 0 d2

  4. Center of Gravity • All forces act as if at the center of mass. • This includes gravity • Sometimes called the center of gravity m1 m2

  5. Symmetry • Continuous objects have a center of mass. • In general the calculus is needed • If an object is regular, the symmetry tells us the center along that direction.

  6. After the split, the sum of momentum is conserved. P = m1v1 + m2v2 Center of mass velocity remains the same. The kinetic energy is not conserved. Before the split, momentum is P = MV M total mass V center of mass velocity Break Up v1 V V M v2

  7. A 325 kg booster rocket and 732 kg satellite coast at 5.22 km/s. Explosive bolts cause a separation in the direction of motion. Satellite moves at 6.69 km/s Booster moves at 1.91 km/s Find the kinetic energy before separation, and the energy of the explosion. Kinetic energy before separation is (1/2)MV2 K = (1/2)(1057 kg)(5.22 x 103 m/s)2 = 14.4 GJ Kinetic energy after separation K1 = 16.4 GJ K2 = 0.592 GJ The difference is the energy of the explosion. Kint = 2.6 GJ Explosions

  8. Thrust • If there is no external force the force to be applied must be proportional to the time rate of change in mass. • The mass changes by Dm • The velocity changes by Dv • The mass added or removed had a velocity u compared to the object The forceu(Dm/Dt) is the thrust

  9. Thrust can be used to find the force of a stream of water. A hose provides a flow of 4.4 kg/s at a speed of 20. m/s. The momentum loss is (20. m/s)(4.4 kg/s) = 88 N The momentum loss is the force. Water Force

  10. Water is poured into a beaker from a height of 2 m at a rate of 4 g/s, into a beaker with a 100. g mass. What does the scale read when the water is at 200. ml in the beaker (1 ml is 1 g)? Answer: 302 g There is extra momentum from the falling water. This is about 0.024 N or an equivalent mass of 2.4 g. Heavy Water next

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