Quiz Samples for Chapter 9 Center of Mass and Linear Momentum

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Grace Perkins
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1 Name: Department: Student ID #: Notice +2 ( 1) points per correct (incorrect) answer No penalty for an unanswered question Fill the blank ( ) with ( ) if the statement is correct (incorrect) : corrections to an incorrect answer : solution Textbook: Walker, Halliday, Resnick, Principles of Physics, Tenth Edition, John Wiley & Sons (2014) 9-1 Center of Mass 1 ( ) where m i and x i are the mass and the position of the ith particle and M = m i is the total mass of the system If we use the vector notation, then r com = 1 M m i r i, where r i is the position vector of the ith particle 2 ( ) The center of mass of a system of particles is the point that moves as though all of the system s mass were concentrated there 3 ( ) The center of mass of a system of particles is the point that moves as though all external forces were applied there 4 ( ) The center of mass of a two-body system is r com = m 1r 1 + m 2 r 2 m 1 + m 2 The center of mass of a system of n particles is defined to be the point whose coordinates are given by x com = 1 M y com = 1 M z com = 1 M m i x i, m i y i, m i z i, 5 (a) ( ) The center of mass of an object must lie within the object The center of mass of a C-shaped object does not lie within the object (b) ( ) All the mass of an object is actually concentrated at its center of mass The center of mass of a system of particles is the point that moves as though all of the system s mass were concentrated there and all external forces were applied there (c) ( ) The center of mass of an object cannot move if there is zero net force on the object If there is zero net force on an object, because of the Newton second law, the acceleration of the object is zero, so the object has constant velocity It does not mean that this object cannot move Page 1 of 5

2 (d) (8) The center of mass of a cylinder must lie on its axis!!!the center of mass of a nonuniform cylinder may not lie on its axis Block A, with a mass of m1, is moving with a velocity of v1 while block B, with a mass of m2, is moving with a velocity of v2 The center of mass of the two-block system is moving with the velocity of 6 ( ) vcom = m1 v1 + m2 v2 4 ( ) Two boys with masses of m1 and m2 stand on a horizontal frictionless surface holding the ends of a light road of length L The boys pull themselves together along the rod When they meet the m1 m2 boy will have moved L 9-2 Newton s Second Law for a System of Particles 1 ( ) Newton s Second Law for a System of Particles: The motion of the center of mass of any system of particles is governed by Newton s second law for a system of particles, which is Fnet = M acom Here Fnet is the net force of all the external forces acting on the system, M is the total mass of the system, and acom is the acceleration of the system s center of mass 2 (a) ( ) The center of mass of a system of particles has a constant velocity if the external forces acting on particles of the system sum to zero (b) ( ) The center of mass of a system of particles remains at the same place if it is initially at rest and the external forces sum to zero At the same instant, a ball of mass m1 is dropped from h above Earth, and a second ball, with a mass of m2, is thrown straight upward from Earth s surface with an initial speed of v They move along nearby q lines and pass without colliding After time t( 2h g ) has elapsed, the velocity of the center of mass of the two-ball system is m2 v gt, where the positive direction is upward 5 ( ) 3 ( ) Page 2 of 5

3 A mass m 1 is attached to one end of a spring with a spring constant k and a mass m 2 is attached to the other end The masses are placed on a horizontal surface and the spring is compressed x The spring is then released with the masses at rest and the masses oscillate When the spring has its equilibrium length for the first time, the m 1 mass has a speed of v The mechanical energy that has been lost to this instant is 9-3 Linear Momentum 1 2 kx2 1 2 m 1v 2 1 m 2 1 v 2 = 0 2 m 2 1 ( ) The linear momentum P of a system of particles is P = Mv com, where v com is the velocity of the system s center of mass and M is the total mass of the system 2 (a) ( ) Two bodies, A and B, have equal kinetic energies The mass of A is α times that of B The ratio of the momentum of A to that of B is 1 : α (b) ( ) Two objects, P and Q, have the same momentum Q can have more kinetic energy than P if it is moving faster than P (c) ( ) A m 1 stone is released from rest and falls toward the Earth After time t has elapsed, the magnitude of its momentum is mgt 3 (a) ( ) Force equals the time rate of change of momentum (b) ( ) If the total momentum of a system is changing, a net external force must be acting on the system (c) ( ) The unit of a momentum may be expressed in N s (d) ( ) The momentum of an object at a given instant is in the same direction as its velocity (e) ( ) When you step on the accelerator to increase the speed of your car, the force that accelerates the car is the force of friction of the road on the tires 9-4 Collision and Impulse 1 ( ) The Impulse Linear Momentum Theorem: The change in the body s linear momentum due to an impulse is p = p f p i = J, where p i and p f are the initial and the final momentum, respectively, and the impulse J is defined by J = tf t i F (t)dt = F average t Here, F average is the time average of the force during the collision of time interval [t i, t f = t i + t] 2 ( ) The physical quantity impulse has the same dimensions as that of momentum 3 (a) ( ) A golf ball of mass m is hit by a golf club so that the ball leaves the tee with speed v The club is in contact with the ball for time T The average force on the club on the ball during the time T is mv/t (b) ( ) A box of books with mass m is dropped on the floor from a height of h and comes to rest The floor exerted impulse of m 2gh on the box 9-5 Conservation of Linear Momentum 1 ( ) If a system is isolated so that no net external force acts on it, the linear momentum of the system remains constant: P = constant P f = P i In any two-body collision of an isolated system, the sum of two momenta is invariant: p 1f + p 2f = p 1i + p 2i Here, 1 and 2 are particle IDs and f and i indicate the initial and final states, respectively In terms of the masses and the velocities, the linear momentum conservation can be stated as m 1 v 1f + m 2 v 2f = m 1 v 1i + m 2 v 2i Page 3 of 5

4 2 ( ) The center of mass of a closed, isolated system of two colliding bodies is not affected by a collision In particular, the velocity of the center of mass cannot be changed by the collision 3 ( ) A shell is launched from a cannon and explodes mid-air The explosion exerts only internal forces on the shell, but gravity is an external force Its horizontal momentum is conserved but its vertical momentum is not conserved 9-6 Momentum and Kinetic Energy in Collisions 1 (a) ( ) A puck of mass m1 is traveling at speed v It strikes another puck with mass m2, which is stationary The two pucks stick together Their common final speed is m1 v 2 (a) ( ) An inelastic collision is one in which momentum is conserved but kinetic energy is not conserved A block moves at speed v in the positive x direction and hits an identical block, initially at rest A small amount of gunpowder had been placed on one of the blocks The explosion does not harm the blocks but it doubles their total kinetic energy After the explosion, the blocks move along the x axis and the incident block has a speed of 1 3 v 2 4 ( ) Cart A, with a mass of m1, travels on a horizontal air track at a speed of v and hits cart B, which has a mass of m2 and is initially at rest After the collision, the center of mass of the two cart system has a speed of m1 v 5 ( ) A bullet of mass m is fired horizontally into a block of wood with mass M suspended by a rope from the ceiling After the bullet sticks into the block, they swing in an arc, rising h above its lowest position The velocity of the bullet was m+mp 2gh m 6 (a) ( ) For a completely inelastic two-body collision the kinetic energy retained by the 2 where M is objects is the same as 21 M vcom the total mass and vcom is the velocity of the center of mass (b) ( ) (b) ( ) An elastic collision is one in which kinetic energy and momentum are both conserved (c) ( ) A completely inelastic collision is one in which the bodies stick together after the collision 3 ( ) A sack of coal with a mass of m is dropped on a railroad flatcar with mass M which was initially moving at v as shown After the sack rests on the flatcar, the speed of the flatcar is M v m+m Page 4 of 5

5 9-7 Elastic Collisions in One Dimension 1 ( ) In an elastic collision, the total kinetic energy is conserved: K1f + K2f = K1i + K2i In addition, the linear momentum is also conserved: p1f + p2f = p1i + p2i 2 ( ) Blocks A and B are moving toward each other along the x axis A has a mass of ma and a velocity of va, while B has a mass of mb and a velocity of vb They suffer an elastic collision and move off along the x axis The velocity of B after the collision is mb vb + 2mA va ma vb ma + mb 9-8 Collisions in Two Dimensions 1 ( ) 2 ( ) A hockey puck of mass m traveling along the x axis at 45 m/s hits another identical hockey puck at rest If after the collision the second puck travels at a speed of 35 m/s at an angle of 30 above the x axis, this is an inelastic collision since kinetic energy is not conserved 9-9 System with Varying Mass: A Rocket 1 ( ) In the absence of external forces a rocket accelerates at an instantaneous rate given by Rvrelative = M a, where R is the fuel consumption rate, vrelative is the speed of fuel ejected relative to the rocket, Rvrelative is the thrust of the rocket engine, M is the variable mass of the rocket The speed changes from vi to vf when its mass changes from Mi to Mf : Mi vf vi = vrelative ln Mf 2 (a) ( ) A rocket exhausts fuel with a velocity of m/s, relative to the rocket It starts from rest in outer space with fuel comprising 80 percent of the total mass When all the fuel has been exhausted its speed is m/s (b) ( ) A hockey puck of mass m traveling along the x axis at v0 hits another identical hockey puck at rest If after the collision the second puck travels at a speed of v2 at an angle of θ2 above the x axis, the final velocity of the first puck is v1, θ1 below the x axis Then θ1 is 1 v0 v2 cos θ2 tan v2 sin θ2 A space probe of mass M is motionless in space To start moving, its main engine is fired for 5 s during which time it ejects exhaust gases at a speed of v At the end of this process, it is moving at V Note that v and V are not relative speed The mass of the ejected gas is V M v+v Page 5 of 5

Momentum Practice Problems

Momentum Practice Problems PSI AP Physics C Name Multiple Choice 1. A steel ball and a piece of clay have equal mass. They are dropped from the same height on a horizontal steel platform. The ball bounces