Orbit calculations using energy and angular momentum conservation
Lecture 31 Angular Momentum; Conservation of Angular Momentum
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Transcript of Lecture 31 Angular Momentum; Conservation of Angular Momentum
Lecture 31: Angular momentum; Conservation of Angular Momentum
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Lecture Objectives: 1. Define angular momentum for a point particle and for
a rigid body. 2. Explain the conditions for conservation of angular
momentum. 3. Apply conservation of angular momentum to physical
systems
(linear) momentum p the larger the momentum, the harder to stop
Newton’s First Law
“an object at rest remains at rest, at object moving will keep on moving… until something (external force) stops it”
Anything that rotates keeps on rotating until something stops it.
Angular momentum
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Angular momentum
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Angular momentum
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The lightweight wheels on racing bikes have less angular momentum than those on recreational bikes, so it takes less effort to get them turning.
Conservation of angular momentum
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Sample Problem An acrobatic physics professor stands at the center of a turntable, holding his arms extended horizontally with a 5.0kg dumbbell in each hand. He is set rotating about a vertical axis, making one revolution in 2.0s. Find the prof’s new angular velocity if he pulls the dumbbells in to his stomach. His moment of inertial (w/o dumbbells) is 3.0kgm2 when his arms are outstretched, dropping to 2.2kgm2 when his hands are in his stomach. The dumbbells are 1.0m from the axis initially and 0.20m from it at the end.
Using conservation of momentum and solving for ω2z :
K2 – K1 = 256J This is extra energy is the work the professor did to pull his arms downward
Two disks: one (A) an engine flywheel and the other (B) a clutch plate attached to a transmission shaft. Their moment of inertia are IA and IB; initially, they are rotating with constant angular speeds ωA and ωB respectively. We then pushed the disks together with force acting along the axis, so as not to apply any torque on either disk. The disk rub against each other and eventually reach a common final angular speed ω. Derive an expression for ω.
Sample Problem: Rotational collision
Suppose flywheel A has mass of 2.0kg, a radius of 0.20m, and an initial angular speed of 50rad/s, and the clutch plate B has mass of 4.0kg, radius of 0.10m and initial angular speed of 200rad/s. Find the common final angular speed ω after the disks are pushed into contact. What happened to the kinetic energy during this process?
Sample Problem: Rotational collision II
The moments of inertia of the two disks are:
From the previous problem, we know the final angular speed, substituting the values:
Calculating the kinetic energy before and after collision:
Therefore: K1 > K2 kinetic energy decreases (just as we would expect from a perfectly inelastic collision ☺)
Seatwork
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Use: L = Iω For a rod at rotating at its end, I = 1/3 MR2
2πrad = 1rev
SW1: Calculate the magnitude of L SW 2: Use right hand rule to get the direction of L
SW 3, 4 and 5
Apply conservation of momentum: Lbefore-‐jump = Lafter-‐jump
Ibar = 1/3Mbarr2
Hint: Just after the bug jumps, it has angular momentum in one direction and the bar is rotating with angular velocity ωB in the opposite direction
1) L = Iω L = 1/3 MR2 ω L = 1/3 (0.006kg)(0.15m)2 (1rev/min) (2πrad/1rev)(1min/60s) L = 4.71x10-‐6kgm/s2
Apply conservation of momentum with the axis at the nail. Let object A be the bug and object B be the bar. Initially, all the objects are at rest and L1 = 0; just after the bug jumps, it has angular momentum in one direction and the bar is rotating with angular velocity ωB in the opposite direction:
Increase in KE comes from the work done by the bug when it pushes against the bar in order to jump ☺