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Section 92 Conservation of Momentum
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Speed A 1325-kg car, C, moving north at 270 m/s, collides with a 2165-kg car, D, moving east at 1 10 m/s The two cars are stuck together In what direction and with what speed do they move after the collision
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Define the system Sketch the before and after states Establish the coordinate axis with the y-axis north and the x-axis east Draw a momentum-vector diagram Known:
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mC mD vCi, y vDi, x 1325 kg 2165 kg 270 m/s 110 m/s
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D vDi 110 m/s Vector Diagram 270 m/s pDi C 90 pCi pf
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Unknown:
vf, x vf, y
Solve for the Unknown
Determine the initial momenta of the cars and the momentum of the system pCi mCvCi, y (1325 kg)(270 m/s) Substitute mC = 1325 kg, vCi, y 4 kg m/s (north) 358 10 pDi mDvDi, x (2165 kg)(110 m/s) Substitute mD 2165 kg, vDi, x 238 104 kg m/s (east) Use the law of conservation of momentum to find pf pf, x pi, x 238 104 kg m/s Substitute pi, x pDi pf, y pi, y 358 104 kg m/s Substitute pi, y pCi Use the diagram to set up equations for pf, x and pf, y pf (pf, x )2 (pf, y )2 (358 104 kg m/s)2 (238 104 kg m/s)2 430 104 kg m/s Solve for tan tan
110 m/s
238 104 kg m/s 358 104 kg m/s
Substitute pf, x 238 104 kg m/s, pf, y 358 104 kg m/s
Math Handbook
( ) (
pf, y pf, x
Inverses of Sine, Cosine, and Tangent page 856
358 104 kg m/s 238 104 kg m/s
Substitute pf, y
358 104 kg m/s, pf, x
238 104 kg m/s
564 Determine the final speed vf
pf (mC 430 104 kg m/s (1325 kg 2165 kg)
430 104 kg m/s, mC
Substitute pf
1325 kg, mD
2165 kg
123 m/s
Evaluate the Answer
Are the units correct The correct unit for speed is m/s Do the signs make sense Answers are both positive and at the appropriate angles Is the magnitude realistic The cars stick together, so vf must be smaller than vCi
9 Momentum and Its Conservation
22 A 925-kg car moving north at 201 m/s collides with a 1865-kg car moving west at 134 m/s The two cars are stuck together In what direction and at what speed do they move after the collision 23 A 1383-kg car moving south at 112 m/s is struck by a 1732-kg car moving east at 313 m/s The cars are stuck together How fast and in what direction do they move immediately after the collision 24 A stationary billiard ball, with a mass of 017 kg, is struck by an identical ball moving at 40 m/s After the collision, the second ball moves 600 to the left of its original direction The stationary ball moves 300 to the right of the moving ball s original direction What is the velocity of each ball after the collision 25 A 1345-kg car moving east at 157 m/s is struck by a 1923-kg car moving north They are stuck together and move with an initial velocity of 145 m/s at = 635 Was the north-moving car exceeding the 201 m/s speed limit
Conservation of Angular Momentum
Like linear momentum, angular momentum can be conserved The law of conservation of angular momentum states that if no net external torque acts on an object, then its angular momentum does not change This is represented by the following equation Law of Conservation of Angular Momentum L1 L2
An object s initial angular momentum is equal to its final angular momentum
Figure 9-1 When the ice1 skater s arms are extended, the moment of inertia increases and his angular velocity decreases (a) When his arms are closer to his body the moment of inertia decreases and results in an increased angular velocity (b)
For example, Earth spins on its axis with no external torques Its angular momentum is constant Thus, Earth s angular momentum is conserved As a result, the length of a day does not change A spinning ice-skater also demonstrates conservation of angular momentum Figure 9-11a shows an ice-skater spinning with his arms extended When he pulls in his arms, as shown in Figure 9-11b, he begins spinning faster Without an external torque, his angular momentum does not change; that is, L I is constant Thus, the ice-skater s increased angular velocity must be accompanied by a decreased moment of inertia By pulling his arms close to his body, the ice-skater brings more mass closer to the axis of rotation, thereby decreasing the radius of rotation and decreasing his moment of inertia You can calculate changes in angular velocity using the law of conservation of angular momentum Li thus, Ii
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