Chapter 5 -- Conservation of Linear Momentum
	Updated ---  18 September 2006
#	Hint	Answer	Update Date
5.1	Parts (a) and (b) are just kinematics. Take a good look at the review exercise on kinematics starting on page 5-5 and you'll be OK. Since this is just a particle, it's easy to calculate it's linear momentum: Psys = mV. And since a particle is a closed system, the conservation of linear momentum equation is nice and simple too.		21Sept05
5.2	FYI --- Most rocket sleds that I know of travel in a horizontal plane. When they travel vertically or leave the ground they are rockets.  So assume that all motion is in a plane perpendicular to the gravity vector. You can probably solve it for vertical motion, but it won't match my answers. (a)  Apply LM and carefully draw an interaction diagram (free-body diagram) carefully locating all force transports of momentum for the rocket sled.. (Assume it is a closed system.) (b)  Again rely on LM equation to find an equation for dV/dt. once you know this use definitions to relate position, velocity, and dV/dt (acceleration). Yup, you do have to integrate. Try finding the general solution (indefinite integration) and then evaluating the constants OR just do a definite integral. Watch out for units they  can help you and hurt you.	(a) k = 16.33 x 10^-3lbf-s^2/ft^2 (b) 78.8 s;  14.1 x 10^3 ft	18Sept06
5.3	Carefully sketch your system boundary because the location of your boundary determines how the forces enter the problem. Is this a steady-state system?	(a) TCA = 1203 N; TCB = 1109 N (c) TCA = 5628 N	17Sept05
5.4	Follow the hints in the problem statement. Be careful with units. Note that in this problem you are using two different systems to study the same process. This is necessary because you are looking for two different pieces of information. This is a very common occurrence in problem solving. Also remember that if you want to know the value of a momentum transport, say a force, then you have to have a boundary at the location where the transport occurs. Be very careful to draw the forces on your free body diagram.	(a) 225.1 ft (b) 4398 lbf in tension	21Sept05
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5.6	Your system should cut the flanges; thus you have both flow into and out of the reducing pipe bend. Carefully apply the conservation of linear momentum to this system paying particular attention to the net pressure force and the net transports of linear momentum with mass flow. Note that mass flow rates across the boundary of a system can carry both positive and negative linear momentum. The plus/minus sign that appears in the rate-form of the conservation of linear momentum equation is there to indicate whether the mass flow rate is into or out of the system. The specific linear momentum V is a vector and also has a sign associated with it.      Ignore the words in parenthesis in the problem. You want to find the net force the bolts would apply to the elbow to hold it in place. Because you are neglecting the weight of the system, your answer will have only an x component.	Fbolts = 14.93 kN to the left	21Sept05
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5.10	Suggest you take the athlete as a closed system. Write rate form of the LM equation for the athlete showing a time varying reaction force of the ground on the athlete. Integrate with respect to time and solve for the average impulsive force, i.e. the constant force which if it acted on the runner during the known time interval would have cause the necessary change in system LM. Giving your answer in terms of the weight of the athlete gives the result in relative terms.	(a) If runner is running to the right, the horizontal force is 1.29Wathlete to the left. (b) The vertical component on the athlete is 6.21Wathlete in a direction opposite to gravity.	21Sept05
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5.13	Two possible systems: Cone alone (a closed system) or enlarged cone-shaped volume (open system) that includes the conical deflector and the air streams. Also note that there are really only TWO inlet/outlets for the second system. The main jet enters from the bottom and then leaves at the top of the deflector in an annular sheet of moving air. Again answer requires application of cons. of LM. QUESTION:  Can you prove that the entering mass flow rate must equal the mass flow rate at the exit? REAL LIFE:  In real life, friction would probably slow down the air velocity; however, as a first approximation assuming no change is velocity is a good model.	m = 0.109 kg	21Sept05
5.14	You will need two very different systems here. For part (a) this should be a particle of sand (a cl___ system) as you travel with it. IS this system steady-state? For part (b) your system should be the region of space icluding the belt and the sand entering and leaving it (an op__ system). OR you could just consider the volume of the sand on the belt as an open, stationary, non-deforming system. Is this a steady-state system? To get the "advanced" answer for part (b), you have to eliminate the mass of the sand in favor of other quantities. How do you do this? Consider looking at conservation of mass for the time before the belt is completely covered with sand.	(a) 6.09� or 52.9� (b) P = msandsin(α) + mdotVo Advanced answer: P = 0.041sin(α)gmdotL + 24mdot With mdot = 100 lbm/s, P is 83.5 lbf with 8.9 lbf from weight of sand and 74.6 lbf from change in LM of sand.	21Sept05
5.15	(1) Recognize that the resistance force can be written as R= kx. (2) Unfortunately this gives us two variables x and t. Change the variables of integration from t to x by using the following substitution dV/dt = (dV/dx)(dx/dt) = V(dV/dx). This type of substitution was discussed Now you can integrate with respect to position since you know something about  V vs x, e.g. V=0 at x = 75 mm. Solve for k using the known velocity-position information. Then you can determine the velocity at x = 25 mm.  Be careful with units.	V25 mm = 565.7 m/s	21Sept05
5.16	Apply conservation of LM to the block.  (What does 125 lb mean?  A mass of 125 lbm or a weight of 125 lbf? Would it really make any difference if process occurs on the earth's surface? IF the process is on the earth and you are assuming standard gravitational acceleration, then your final answer will be the same regardless of which interpretation you assume.  Try it!) (a)  Remember the block will not move until the applied force just exceeds the maximum static friction force. (b)  How do you usually find the maximum or minimum of a function? (b)  After it starts moving the friction force drops to the kinetic friction value. You will probably have to integrate the LM equation with time over several time intervals to handle the varying forces.	(a) t = 5 s (b) t = 12 s;  Vmax = 49.9 ft/s (c) t = 17.88 s	24Sept07
5.17	For part (a) you must find out if the boxes move together or if they pull apart.  To do this examine each block as if it was alone on the conveyor belt. IF the acceleration of Block A, all alone, is greater than the acceleration of Block A, all alone, THEN Block A pushes Block B and they move together as a single unit. IF the converse is true, then Block B will pull away from Block A and they will separate. There is only one answer to part (b), depending upon what you find in part (a). If they move as a single unit, you must find the actual acceleration of the combined system with the appropriate friction forces for each box.	aa = 0.997 ft/s^2	21Sept05
5.18	As is the case with many problems involving friction, you do not know what the state of motion is a priori. You must therefore assume some state of motion first, apply CoLM, and then chack that assumption against something else. I assumed that block A did not slide relative to B, solved CoLM for the friciton force in between the blocks, and then compared that value to the critical value of friction when impending motion would occur (which is...) You must use at least two systems here. Just one is not enough. And if you screw up your free body diagram, there is no hope. (Assume the pulley is weightless and fricitonless so that the force P has the same magnitude wherever your system boundary cuts through it.)	(a) No sliding. |aA| = |aB| = 0.667 m/s^2 (c) Sliding. |aA| = 5.1 m/s^2, |aB| = 0.980 m/s^2	21Sept05
5.19	(a) Application of finite-time form of the cons. of LM to the package and cart combined. Be sure to show all forces that act on your system when you draw your system diagram. (If both cart and package are in the system are there any forces drawn between the package and cart? NO! Why?) (b) Since you want impulse exerted on package by the cart, you want to find the time integral of the forces that the cart exerts on the package during the impact. To do this you must select the system as your package. The x-component is independent of the time interval; however, the vertical component of the impulse depends on the time for the impact. IF Dt << 1 second, the impulse in the positive y-direction is 15.00 N-s.	(a)  0.742 m/s (b) RxavgDt = -18.56 N-s RyavgDt = (15.00 N-s) + (98.1 N)Dt where are forces on the package and point to the right and up respectively	21Sept05
5.20	Apply conservation of LM to combined system of both cars, then solve for the ratio of Va to Vb.  This will tell you who is going faster, then solve B. Slower car is traveling at the speed limit of 50 km/h.  Don't forget that LM has two components in the x-y plane. Gravity plays no role in this problem.	(b)  VA = 192 km/hr	21Sept05
5.21	To find the tension in the rope, where should your system boundary be? What information have you been given by looking at the speed reduction over time?.	(a) 8.51 km/h (b) 6.67 N	21Sept05
5.22	Clarifications:  Pressure at the inlet to the elbow is 2 psig (or 16.7 psia) The pressure at the exit of the elbow is atmospheric pressure, 0 psig (or 14.7 psia.)  Pressures reported as psia are measured with respect to a complete vacuum; pressures reported as psig are measured with respect to the local pressure of the atmosphere. The abbreviation psig psig stands for "pounds per square inch gauge" which is a way of reporting pressures relative to atmospheric pressure, i.e. 2 psig = (2 + 14.7) psia  "pounds per square inch absolute."  The KEY issue is that all pressures are reported in the same way for purposes of the calculation, i.e. all in psig or all in psia. When atmospheric pressure acts over part of a system, the easiest way is to account for the net pressure forces is to use all pressures in psig. Then the only those pressures that differ from atmospheric will produce a net force. [Note that it is impossible to have negative absolute pressures; however, a vacuum will have a negative gage pressure, e.g. 12.7 psia = -2.0 psig.]	---Horizontal force acting on the elbow by the pipe = 160.9 lbf to the right ---Vertical force acting on the elbow by the pipe = 122.3  lbf down.	21Sept05
5.23	Part (a): the impact is so short, the impulse do to friction can be neglected. Part (b): assume the blocks immediately start to slip relative to each other (you will want to use two systems due to the friction between the blocks). Part (c): when the blocks stop sliding on each other, they will have equal velocities and accelerations relative to the ground.	a) v = 0.86 m/s (right) b) aA = 0.97 m/s^2 (left), aC = 2.94 m/s^2 (right) c) t = 0.22s
5.24	Determine the unknown constant A from the given information. Then answer the other questions as required.  Suggest you use a spreadsheet or MAPLE to graph the various quantities to make life simple for your self.	(b)  168.5 m	21Sept05
5.25	Apply conservation of linear momentum to the system and recognize that there is no motion; thus, the system has a constant value of linear momentum.	TAC= 1376 N; TAB= 1601 N	18Sept06
5.26	Apply conservation of LM to the block. Although any coordinate system will work, the easiest coordinate system to use is one aligned with the inclined plane, i.e. x is parallel to the inclined plane and y is perpendicular to the plane.     Biggest wrinkle here is that the force P also contributes to the normal force Fn of the plane acting on the block.  Don't forget to consider that the block could move by sliding up the plane or down the plane. Be very careful with angles.  Also be careful with the direction of the friction force when the motion is impending	(a)  56.97 N  to 318.6 N (b)  ---	21Sept05
5.27	(a) Carefully consider your system. If you pick both the block A and mass B as the system, you will have two unknowns: dV/dt and F. Thus you will need an additional relation between these two variables or know at least one of them. Is there another system, you can pick. [Hint: Consider just the mass and note that the link can only transmit a force that is parallel to the link.]	(a) 207.4 N	21Sept05
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5.32	Do not be scared by the lack of numbers, just follow the standard COM and COLM method. Note, the drag force pulls the boat to the right.	a) v1 = V_dot/A1, v2 = V_dot/A2 b) V_dot = vw*sqrt((kA1A2)/(rho(A1-A2cos(theta))))
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5.37	Remember that velocity has a direction; this is important for the m_dot*v terms. Also consider which inlet or outlet has an associated gauge/vacuum pressure; if one exists you will need to include it on your FBD.	R = 34.9 kN (down)
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5.40	Maximum deceleration can occur only when the maximum friction force is developed.	dv/dt = 1.25 m/s^2 (left)
5.41	If block B immediately moves, you know kinetic friction must be involved. Since there is friction between the blocks, it will be easiest to use two systems.	T = 97.7 N aB = 2.93 m/s (down the ramp)
5.42	What exactly are you trying to find here? Would it make sense to treat the car and the ball as a single system? Consider the ball as a closed system. Use LM to relate the projectile trajectory to its initial velocity. Note that the LM of the ball must be written in terms of its absolute velocity measured with respect to the ground not its relative velocity with respect to the car (the given information in the problem). Use relative velocity relations to relate the initial ball velocity required for the given trajectory to the the  given information and the car velocity.	(a) 0.792 s (b)  44.9 ft/s	21Sept05
5.43	Apply both conservation of mass and conservation of linear momentum to an open system that corresponds to the cart and water in the tank. Assume that all of the water in the tank is moving with the velocity of the cart. Take care when you differentiate the LM of the open system and use the cons. of mass where applicable to relate dm/dt to the mass flow rate. Also note that the mass flow rate must be calculated with a velocity that is measured relative to the flow area.	vc,max = (v0 - F/(rhov0A)(ln(mc+mw)/mc)	26Sept05
5.44	Use a closed system that minimizes the unknown information. Use the relative velocity relations to relate the absolute velocities of the boat and swimmer(s) to the absolute velocity of the swimmer(s).	(a) 1.154 m/s ← (b) 1.292 m/s ← (c)  1.280 m/s ←	29Sept05