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Physics 7C lecture 09 2D collision, center of mass Tuesday October 29, 8:00 AM – 9:20 AM Engineering Hall 1200 Copyright © 2012 Pearson Education Inc. Review: Momentum and Newton’s second law • The momentum of a particle is the product of its mass and its velocity: • Newton’s second law can be written in terms of momentum as Copyright © 2012 Pearson Education Inc. Impulse and momentum • The impulse of a force is the product of the force and the time interval during which it acts. • On a graph of Fx versus time, the impulse is equal to the area under the curve, as shown in Figure 8.3 to the right. • Impulse-momentum theorem: The change in momentum of a particle during a time interval is equal to the impulse of the net force acting on the particle during that interval. Copyright © 2012 Pearson Education Inc. Conservation of momentum • External forces (the normal force and gravity) act on the skaters shown in Figure 8.9 at the right, but their vector sum is zero. Therefore the total momentum of the skaters is conserved. • Conservation of momentum: If the vector sum of the external forces on a system is zero, the total momentum of the system is constant. Copyright © 2012 Pearson Education Inc. Objects colliding along a straight line • Two gliders collide on an air track in Example 8.5. • Follow Example 8.5 using Figure 8.12 as shown below. Copyright © 2012 Pearson Education Inc. Elastic collisions •In an elastic collision, the total kinetic energy of the system is the same after the collision as before. Copyright © 2012 Pearson Education Inc. Q8.8 An open cart is rolling to the left on a horizontal surface. A package slides down a chute and lands in the cart. Which quantities have the same value just before and just after the package lands in the cart? A. the horizontal component of total momentum B. the vertical component of total momentum C. the total kinetic energy D. two of A., B., and C. E. all of A., B., and C. Copyright © 2012 Pearson Education Inc. A8.8 An open cart is rolling to the left on a horizontal surface. A package slides down a chute and lands in the cart. Which quantities have the same value just before and just after the package lands in the cart? A. the horizontal component of total momentum B. the vertical component of total momentum C. the total kinetic energy D. two of A., B., and C. E. all of A., B., and C. Copyright © 2012 Pearson Education Inc. A two-dimensional collision • Two robots collide and go off at different angles. • If the angles are known, find out final speeds. Copyright © 2012 Pearson Education Inc. A two-dimensional collision • momentum conservation along x and y. Copyright © 2012 Pearson Education Inc. An automobile collision • Two cars traveling at right angles collide into each other, find out the final speed and direction. Copyright © 2012 Pearson Education Inc. An automobile collision momentum conservation: Copyright © 2012 Pearson Education Inc. A two-dimensional elastic collision • find out α, β and vb2 after elastic collision. Copyright © 2012 Pearson Education Inc. A two-dimensional elastic collision Copyright © 2012 Pearson Education Inc. Center of mass • We have talked about the motion of cars, planets etc. as if they are a single point. Why can we do this? • What is unique about the white point in the falling wrench? Copyright © 2012 Pearson Education Inc. Center of mass • The center of mass, of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero Copyright © 2012 Pearson Education Inc. Center of mass • for a continuous volume Copyright © 2012 Pearson Education Inc. Center of mass of a water molecule • Where is the center of mass of a water molecule? Copyright © 2012 Pearson Education Inc. Center of mass of a water molecule • Where is the center of mass of a water molecule? Copyright © 2012 Pearson Education Inc. Q8.9 A yellow block and a red rod are joined together. Each object is of uniform density. The center of mass of the combined object is at the position shown by the black “X.” Which has the greater mass, the yellow block or the red rod? A. the yellow block B. the red rod C. They both have the same mass. D. not enough information given to decide Copyright © 2012 Pearson Education Inc. A8.9 A yellow block and a red rod are joined together. Each object is of uniform density. The center of mass of the combined object is at the position shown by the black “X.” Which has the greater mass, the yellow block or the red rod? A. the yellow block B. the red rod C. They both have the same mass. D. not enough information given to decide Copyright © 2012 Pearson Education Inc. Center of mass of symmetrical objects • It is easy to find the center of mass of a homogeneous symmetric object. Copyright © 2012 Pearson Education Inc. Motion of the center of mass • The total momentum of a system is equal to the total mass times the velocity of the center of mass. • The center of mass of the wrench moves as though all the mass were concentrated there. Copyright © 2012 Pearson Education Inc. Tug-of-war on the ice • When James moved 6 m toward the mug, how far has Ramon moved? Copyright © 2012 Pearson Education Inc. Tug-of-war on the ice • Using momentum conservation: Copyright © 2012 Pearson Education Inc. Tug-of-war on the ice • Using motion of center of mass: Copyright © 2012 Pearson Education Inc. 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. Copyright © 2012 Pearson Education Inc. 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. Copyright © 2012 Pearson Education Inc. 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: Copyright © 2012 Pearson Education Inc. 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? Copyright © 2012 Pearson Education Inc. 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) Copyright © 2012 Pearson Education Inc. 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 –vex λ dt = 0 dv - λ vex dt /(m0- λ t)= 0 v + vex ln(m0- λ t) = constant v = v0 + vex ln (m0/(m0- λ t)) = v0 + vex ln (m0/m) Copyright © 2012 Pearson Education Inc.