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Physics 7C lecture 10. Rotation. Thursday October 31, 8:00 AM – 9:20 AM Engineering Hall 1200. External forces and center-of-mass motion. When a body or collection of particles is acted upon by external forces, the center of mass moves as though all the mass were concentrated there.

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**Physics 7C lecture 10**• Rotation Thursday October 31, 8:00 AM – 9:20 AM Engineering Hall 1200**External forces and center-of-mass motion**• When a body or collection of particles is acted upon by external forces, the center of mass moves as though all the mass were concentrated there.**External forces and center-of-mass motion**• Fragments of a firework shell would fly at 100 m/s for 5 seconds before they burn out. If a shell reaches its max height of 1000 meter and explodes, are the audiences on the ground safe from burning fragments? Ignore air resistance.**External forces and center-of-mass motion**• Fragments of a firework shell would fly at 100 m/s for 5 seconds before they burn out. If a shell reaches its max height of 1000 meter and explodes, are the audiences on the ground safe from burning fragments? Ignore air resistance. motion of center of mass: motion of fragments relative to center of mass:**Rocket propulsion**• As a rocket burns fuel, its mass decreases, as shown in Figure below. • What is the speed of rocket if we know the exhaust speed vex, burning rate λ=dm/dt and initial mass m0?**Rocket propulsion**• What is the speed of rocket if we know the exhaust speed vex, burning rate λ=dm/dt and initial mass m0? between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm(v-vex)**Rocket propulsion**• What is the speed of rocket if we know the exhaust speed vex, burning rate λ=dm/dt and initial mass m0? between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm(v-vex) m dv – v dm+ (v-vex) dm= 0 (m0- λ t) dv –vexλdt = 0 dv - λvex dt /(m0- λ t)= 0 v + vex ln(m0- λ t) = constant v = v0 + vexln (m0/(m0- λ t)) = v0 + vexln (m0/m)**Introduction**• The north star is Polaris today, but 5000 years ago it was Thuban. What caused the change? • What causes bodies to start or stop spinning? • We’ll introduce some new concepts, such as torque and angular momentum, to deepen our understanding of rotational motion.**Introduction**• How do we quantify the spinning of wind turbine?**Angular displacement**• angular displacement: θ • unit: radian • direction: (right hand rule!)**Radian**2 π radian = 360 degree**Angular displacement**• Motion of a spinning wheel**Angular displacement and velocity**• How do we quantify the spinning of wind turbine? ω = dθ /dt**Angular velocity is a vector!**• signs of angular displacement**Right hand rule**• Angular displacement is a vector, use right hand rule to determine the direction.**Angular acceleration**• α = dω/dt**Angular acceleration**• calculate ω from α**These are very similar to linear motion**• linear and angular motion:**Linear vs. angular motion**• v =?**Linear vs. angular motion**• what is the acceleration?**Linear vs. angular motion**• radian vs. degree**Example**• calculate the acceleration of the black point in the disk.**Example**• calculate the acceleration of the black point in the disk.**Example**• calculate the acceleration of the black point in the disk.**Speed of propeller**• calculate the speed of the tip of the propeller.**Speed of propeller**• calculate the speed of the tip of the propeller.

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