Collisions & Center of Mass Lecturer: Professor Stephen T. Thornton

Reading Quiz The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ? A) higher B) lower C) at the same place there is no definable CM in this case (1) X CM (2) Answer: 1

Reading Quiz The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ? A) higher B) lower at the same place there is no definable CM in this case The CM of each half is closer to the top of the semicircle than the bottom. The CM of the whole system is located at the midpoint of the two semicircle CMs, which is higher than the yellow line. (1) X CM (2) CM

Last Time Defined linear momentum Relationship between K.E. and momentum More general form of 2nd law Impulse Internal and external forces

Today Collisions – elastic, inelastic, perfectly inelastic Center of mass Changing mass - rockets

Collisions Momentum is conserved. *** When two or more objects strike each other, we have a collision. Consider for now only two objects. Let the net external force = 0. Internal forces are not considered. Momentum is conserved. *** In general, kinetic energy is not conserved. ***

Kinds of Collisions Elastic collision – kinetic energy (K or KE) is conserved. (definition of elastic collision.) Inelastic collision – KE not conserved due to lost energy (heat, sound, deformation, etc.) Perfectly inelastic collision – objects coalesce together. Do demo with air track cars. Try to show on air track. Hard to do inelastic collision – maybe bang the two carts together.

Remember that linear momentum is always conserved in all collisions with no external force – elastic or inelastic. One dimensional collisions are easiest to consider. In two dimensions, linear momentum is conserved in both directions. It is a vector.

Collisions in Two or Three Dimensions Conservation of energy and momentum can also be used to analyze collisions in two or three dimensions, but unless the situation is very simple, the math quickly becomes unwieldy. Here, a moving object collides with an object initially at rest. Knowing the masses and initial velocities is not enough; we need to know the angles as well in order to find the final velocities. Figure 9-18. Caption: Object A, the projectile, collides with object B, the target. After the collision, they move off with momenta pA’ and pB’ at angles θA’ and θB’. The objects are shown here as particles, as we would visualize them in atomic or nuclear physics. But they could also be macroscopic pool balls.

Conceptual Quiz: After a collision between a large truck and a small car, the impulse given to the truck by the car is   A)  larger than that given to the car by the truck. B)  smaller than that given to the car by the truck. C)  equal to that given to the car by the truck. D)  dependent upon the collision being elastic or inelastic.

Answer: C - they are equal Remember that impulse is equal to the change in linear momentum. There are no external forces. Linear momentum is conserved.

An Elastic Collision Between Two Air Carts Show this on air track.

Elastic Collisions Between Air Carts of Various Masses Equal mass Do on air track.

Elastic Collisions Between Air Carts of Various Masses Do on air track.

Elastic Collisions Between Air Carts of Various Masses Do on air track.

Relative velocities in head-on elastic collisions are equal in magnitude but opposite in direction. Used often in problems. Easier than conserving kinetic energy.

Projectile Explosion. A 224-kg projectile, fired with a speed of 116 m/s at a 60.0° angle, breaks into three pieces of equal mass at the highest point of its arc (where its velocity is horizontal). Two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion; one of these moves vertically downward and the other horizontally. Determine (a) the velocity of the third fragment immediately after the explosion and (b) the energy released in the explosion. Giancoli, 4th ed, Problem 9-21

Center of Mass The center of mass is special because the object acts in many cases as if all the mass of an object is concentrated at that point. The center of mass of an object is the point where the system is balanced in a uniform gravitational field.

The Center of Mass of Two Objects No rotation!

Center of Mass of Two Objects Note that The center of mass is clearly nearest the heavier mass.

It does not matter where you place your coordinate system. Center of mass of three point-like objects: It does not matter where you place your coordinate system.

Center of Mass of the Arm It does not matter where you place your coordinate system.

For an extended object, we imagine making it up of tiny particles, each of tiny mass, and adding up the product of each particle’s mass with its position and dividing by the total mass. In the limit that the particles become infinitely small, this gives: Figure 9-26. Caption: An extended object, here shown in only two dimensions, can be considered to be made up of many tiny particles (n), each having a mass Δmi. One such particle is shown located at a point ri = xii + yij + zik. We take the limit of n →∞ so Δmi becomes the infinitesimal dm.

The center of gravity is the point at which the gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object. Figure 9-30. Caption: Determining the CM of a flat uniform body.

Do center of mass demos Center of mass of Virginia Toys

Motion of Center of Mass Velocity of center of mass: Acceleration of center of mass: Newton’s 2nd Law for System The CM of a system of particles with total mass M moves as if M is acted upon by the net, external force.

More on Center of Mass Motion Note if there is no net external force, then the acceleration of the center of mass is zero.

More on Center of Mass Motion If there is a net external force, the CM moves as if all the net force were acting at the center of mass. Flip a meter stick or ruler to show this.

Crash of Air Carts Note motion of CM.

Center of Mass of an Exploding Rocket

Conceptual Quiz: The object is suspended from two points as shown Conceptual Quiz: The object is suspended from two points as shown. Which number lies closest to the center of mass? Use A, B, C… for 1, 2, 3…

Answer: D #4 Look and see which number is always on a straight vertical line from the point of hanging. It helps to draw a line.

Do ballistic pendulum demo and calculate v0. X should be about 10 cm to obtain 250-350 m/s muzzle velocity m = 2.00 g M = 3.81 kg L = 4.00 m

The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM. Figure 9-22. Caption: Translation plus rotation: a wrench moving over a horizontal surface. The CM, marked with a red cross, moves in a straight line.

Systems of Variable Mass; Rocket Propulsion Applying Newton’s second law to the system shown gives: or The Fext is gravity and air resistance. In the rocket problem, dM/dt is negative (mass is expelled), and so is vrel. This last term is the thrust and this is what causes the rocket to go forward.

Conceptual Quiz vi M vf A) 10 m/s B) 20 m/s C) 0 m/s D) 15 m/s E) 5 m/s A box slides with initial velocity 10 m/s on a frictionless surface and collides inelastically with an identical box. The boxes stick together after the collision. What is the final velocity? vf vi M Click to add notes

Conceptual Quiz vi M vf A) 10 m/s B) 20 m/s C) 0 m/s D) 15 m/s E) 5 m/s A box slides with initial velocity 10 m/s on a frictionless surface and collides inelastically with an identical box. The boxes stick together after the collision. What is the final velocity? The initial momentum is: M vi = (10) M vf vi M The final momentum must be the same!! The final momentum is: Mtot vf = (2M) vf = (2M) (5)

Conceptual Quiz A) the car B) the truck C) they both have the same momentum change D) can’t tell without knowing the final velocities A small car and a large truck collide head-on and stick together. Which one has the larger momentum change? Click to add notes

Conceptual Quiz A) the car B) the truck C) they both have the same momentum change D) can’t tell without knowing the final velocities A small car and a large truck collide head-on and stick together. Which one has the larger momentum change? Because the total momentum of the system is conserved, that means that Dp = 0 for the car and truck combined. Therefore, Dpcar must be equal and opposite to that of the truck (–Dptruck) in order for the total momentum change to be zero. Note that this conclusion also follows from Newton’s Third Law. Follow-up: Which one feels the larger acceleration?

Conceptual Quiz A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater momentum? A) the heavy one B) the light one both have the same momentum D) impossible to say 1 2 Answer: 3

Conceptual Quiz A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the greater momentum? A) the heavy one B) the light one both have the same momentum D) impossible to say 1 2 The initial momentum of the uranium was zero, so the final total momentum of the two fragments must also be zero. Thus the individual momenta are equal in magnitude and opposite in direction.

Conceptual Quiz A) only I B) only II C) I and II D) II and III E) all three If all three collisions below are totally inelastic, which one(s) will bring the car on the left to a complete halt? Answer: 5

Conceptual Quiz A) I B) II C) I and II D) II and III E) all three If all three collisions below are totally inelastic, which one(s) will bring the car on the left to a complete halt? In case I, the solid wall clearly stops the car. In cases II and III, because ptot = 0 before the collision, then ptot must also be zero after the collision, which means that the car comes to a halt in all three cases.