Chapter 6

A fellow student states that the total momentum of a three-particle system \(\left(m_{1}=0.25 \mathrm{~kg}, m_{2}=0.20 \mathrm{~kg},\right.\) and \(m_{3}=0.33 \mathrm{~kg}\) ) is initially zero. He calculates that after an inelastic triple collision the particles have velocities of \(4.0 \mathrm{~m} / \mathrm{s}\) at \(0^{\circ}, 6.0 \mathrm{~m}\) at \(120^{\circ},\) and \(2.5 \mathrm{~m} / \mathrm{s}\) at \(230^{\circ},\) respec- tively, with angles measured from the \(+x\) -axis. Do you agree with his calculations? If not, assuming the first two answers to be correct, what should be the momentum of the third particle so the total momentum is zero?

A freight car with a mass of \(25000 \mathrm{~kg}\) rolls down an inclined track through a vertical distance of \(1.5 \mathrm{~m}\). At the bottom of the incline, on a level track, the car collides and couples with an identical freight car that was at rest. What percentage of the initial kinetic energy is lost in the collision

In nuclear reactors, subatomic particles called neutrons are slowed down by allowing them to collide with the atoms of a moderator material, such as carbon atoms, which are 12 times as massive as neutrons. (a) In a head-on elastic collision with a carbon atom, what percentage of a neutron's energy is lost? (b) If the neutron has an initial speed of \(1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}\), what will be its speed after collision?

In a noninjury chain-reaction accident on a foggy freeway, car 1 (mass of \(2000 \mathrm{~kg}\) ) moving at \(15.0 \mathrm{~m} / \mathrm{s}\) to the right elastically collides with car 2 , initially at rest. The mass of car 2 is \(1500 \mathrm{~kg}\). In turn, car 2 then goes on to lock bumpers (that is, it is a completely inelastic collision) with car \(3,\) which has a mass of \(2500 \mathrm{~kg}\) and was also at rest. Determine the speed of all cars immediately after this unfortunate accident.

(a) The center of mass of a system consisting of two 0.10-kg particles is located at the origin. If one of the particles is at \((0,0.45 \mathrm{~m}),\) where is the other? \((\mathrm{b})\) If the masses are moved so their center of mass is located at \((0.25 \mathrm{~m}\), \(0.15 \mathrm{~m}\) ), can you tell where the particles are located?

Find the center of mass of a system composed of three spherical objects with masses of \(3.0 \mathrm{~kg}, 2.0 \mathrm{~kg},\) and \(4.0 \mathrm{~kg}\) and centers located at \((-6.0 \mathrm{~m}, 0),(1.0 \mathrm{~m}, 0),\) and \((3.0 \mathrm{~m}, 0),\) respectively.

A \(3.0-\mathrm{kg}\) rod of length \(5.0 \mathrm{~m}\) has at opposite ends point masses of \(4.0 \mathrm{~kg}\) and \(6.0 \mathrm{~kg}\). (a) Will the center of mass of this system be (1) nearer to the \(4.0-\mathrm{kg}\) mass, (2) nearer to the 6.0 -kg mass, or (3) at the center of the rod? Why? (b) Where is the center of mass of the system?

A piece of uniform sheet metal measures \(25 \mathrm{~cm}\) by \(25 \mathrm{~cm}\). If a circular piece with a radius of \(5.0 \mathrm{~cm}\) is cut from the center of the sheet, where is the sheet's center of mass now?

Two cups are placed on a uniform board that is balanced on a cylinder ( \(\mathbf{v}\) Fig. 6.40 ). The board has a mass of \(2.00 \mathrm{~kg}\) and is \(2.00 \mathrm{~m}\) long. The mass of \(\operatorname{cup} 1\) is \(200 \mathrm{~g}\) and it is placed \(1.05 \mathrm{~m}\) to the left of the balance point. The mass of cup 2 is \(400 \mathrm{~g}\). Where should cup 2 be placed for balance (relative to the right end of the board)?

Two skaters with masses of \(65 \mathrm{~kg}\) and \(45 \mathrm{~kg}\), respectively, stand \(8.0 \mathrm{~m}\) apart, each holding one end of a piece of rope. (a) If they pull themselves along the rope until they meet, how far does each skater travel? (Neglect friction.) (b) If only the 45 -kg skater pulls along the rope until she meets her friend (who just holds onto the rope), how far does each skater travel?

Three particles, each with a mass of \(0.25 \mathrm{~kg},\) are located at \((-4.0 \mathrm{~m}, 0),(2.0 \mathrm{~m}, 0),\) and \((0,3.0 \mathrm{~m})\) and are acted on by forces \(\overrightarrow{\mathbf{F}}_{1}=(-3.0 \mathrm{~N}) \hat{\mathbf{y}}, \overrightarrow{\mathbf{F}}_{2}=(5.0 \mathrm{~N}) \hat{\mathbf{y}},\) and \(\overrightarrow{\mathbf{F}}_{3}=(4.0 \mathrm{~N}) \hat{\mathbf{x}},\) respectively. Find the acceleration \((\mathrm{mag}-\) nitude and direction) of the center of mass of the system. [Hint: Consider the components of the acceleration.]

A \(170-\mathrm{g}\) hockey puck sliding on ice perpendicularly impacts a flat piece of sideboard. Its incoming momentum is \(6.10 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). It rebounds along its incoming path after having suffered a momentum change (magnitude) of \(8.80 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). (a) If the impact with the board took \(35.0 \mathrm{~ms}\), determine the average force (including direction) exerted by the puck on the board. (b) Determine the final momentum of the puck. (c) Was this collision elastic or inelastic? Prove your answer mathematically.

You are traveling north and make a \(90^{\circ}\) right-hand turn east on a flat road while driving a car that has a total weight of 3600 lb. Before the turn, the car was traveling at \(40 \mathrm{mi} / \mathrm{h},\) and after the turn is completed you have slowed to \(30 \mathrm{mi} / \mathrm{h}\). If the turn took \(4.25 \mathrm{~s}\) to complete, determine the following: (a) the car's change in kinetic energy, (b) the car's change in momentum (including direction), and (c) the average net force exerted on the car during the turn (including direction).

In the radioactive decay of a nucleus of an atom called americium- 241 (symbol \({ }^{241} \mathrm{Am}\), mass of \(\left.4.03 \times 10^{-25} \mathrm{~kg}\right)\), it emits an alpha particle (designated as \(\alpha\) ) with a mass of \(6.68 \times 10^{-27} \mathrm{~kg}\) to the right with a kinetic energy of \(8.64 \times 10^{-13} \mathrm{~J}\). (This is typical of nuclear energies, small on the everyday scale.) The remaining nucleus is neptunium- \(237\left({ }^{237} \mathrm{~Np}\right)\) and has a mass of \(3.96 \times 10^{-25} \mathrm{~kg}\). Assume the initial nucleus was at rest. (a) Will the neptunium nucleus have (1) more, (2) less, or (3) the same amount of kinetic energy compared to the alpha particle? (b) Determine the kinetic energy of the \({ }^{23}\) Np nucleus afterward.

A young hockey player with a mass of \(30.0 \mathrm{~kg}\) is initially moving at \(2.00 \mathrm{~m} / \mathrm{s}\) to the east. He intercepts and catches on the stick a puck initially moving at \(35.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(\theta=60^{\circ}\) (vFig. 6.41). Assume that the puck's mass is \(0.180 \mathrm{~kg}\) and the player and puck form a single object for a few seconds. (a) Determine the direction angle and speed of the puck and skater after the collision. (b) Was this collision elastic or inelastic? Prove your answer with numbers.

In a laboratory setup, two frictionless carts are placed on a horizontal surface. Cart A has a mass of \(500 \mathrm{~g}\) and cart B's mass is \(1000 \mathrm{~g}\). Between them is placed an ideal (very light) spring and they are squeezed together carefully, thereby compressing the spring by \(5.50 \mathrm{~cm} .\) Both carts are then released and \(\mathrm{B}^{\prime}\) s recoil speed is measured to be \(0.55 \mathrm{~m} / \mathrm{s}\). (a) Will cart A's speed be (1) greater than, (2) less than, or (3) the same as B's speed? Explain. (b) Determine B's recoil speed to see if your conjecture in (a) was correct. (c) Determine the spring constant of the spring.