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that is nearly elastic is that of two steel blocks on ice. Another nearly elastic collision is that between two carts with spring bumpers on an air track. Icy surfaces and air tracks are nearly frictionless, more readily allowing nearly elastic collisions on them. Elastic Collision An elastic collision is one that con... |
the system before the collision: ( + ) = + (3.0) = (1)(12) + (2)( − 12) = − 4.0 m/s (8.46) (8.47) (8.48) 330 Chapter 8 | Linear Momentum and Collisions After the collision, the center-of-mass velocity is the same: ( + ) = ( + ) (3.0) = (3)( − 4.0) = − 4.0 m/s The total momentum of the system before the collision is: +... |
, Solving the first equation (momentum equation) for ′2, we obtain 1 21 1 2 = 1 21 ′1 2 + 1 22 ′2 2. ′2 = 1 2 1 − ′1. (8.55) (8.56) (8.57) (8.58) Substituting this expression into the second equation (internal kinetic energy equation) eliminates the variable ′2, leaving only ′1 as an unknown (the algebra is left as an ... |
es and Elastic Collision Find a few ice cubes which are about the same size and a smooth kitchen tabletop or a table with a glass top. Place the ice cubes on the surface several centimeters away from each other. Flick one ice cube toward a stationary ice cube and observe the path and velocities of the ice cubes after t... |
) • 5.D.2.1 The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic. (S.P. 6.4, 7.2) • 5.D.2.2 The student is able to plan data collection strategies to test the la... |
astic Collision An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). Figure 8.11 shows an example of an inelastic collision. Two objects that have equal masses head toward one another at equal speeds and then stick together. Their total internal kinetic energy is initially 1... |
that the final velocity of the puck and goalie are the same. Once the final velocity is found, the kinetic energies can be calculated before and after the collision and compared as requested. Solution for (a) Momentum is conserved because the net external force on the puck-goalie system is zero. Conservation of moment... |
internal kinetic energy during a collision. Figure 8.13 shows a one-dimensional example in which two carts on an air track collide, releasing potential energy from a compressed spring. Example 8.6 deals with data from such a collision. Figure 8.13 An air track is nearly frictionless, so that momentum is conserved. Mot... |
Determine for the cases in Part 1 and where is the height to This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 | Linear Momentum and Collisions 335 for the case of a tennis ball bouncing off a concrete or wooden floor ( = 0.85 for new tennis balls used on a tennis court). Example 8.6... |
kg ( – 0.500 m/s)2 After the collision, the internal kinetic energy is = 0.763 J. 2 + 1 KE′int = 1 = 1 2 = 6.22 J. 21 ′1 0.350 kg 2 22 ′2 (-4.00 m/s)2 + 1 2 0.500 kg (3.70 m/s)2 The change in internal kinetic energy is thus KE′int − KEint = 6.22 J − 0.763 J = 5.46 J. (8.72) (8.73) (8.74) (8.75) Discussion The final ve... |
. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously. One complication a... |
http://cnx.org/content/col11844/1.13 Chapter 8 | Linear Momentum and Collisions 337 The components of the velocities along the -axis have the form cos. Because particle 1 initially moves along the -axis, we find 1 = 1. Conservation of momentum along the -axis gives the following equation: 1 1 = 1′1 cos 1 + 2′2 cos 2, ... |
the velocity of the center-of-mass of this system before and after an inelastic collision, in which the cars move together as one mass after the collision? Since both cars have equal mass, the center-of-mass velocity components are just the average of the components of the individual velocities before the collision. T... |
′2 cos 2 for 2′ cos 2 and 0 = 1′1 sin 1 + 2′2 sin 2 for ′2 sin 2 and taking the ratio yields an equation (in which θ2 is the only unknown quantity. Applying the identity tan = sin cos, we obtain: tan 2 = ′1 sin 1 ′1 cos 1 − 1. Entering known values into the previous equation gives tan 2 = (1.50 m/s)(0.7071) (1.50 m/s)(... |
. Elastic Collisions of Two Objects with Equal Mass Some interesting situations arise when the two colliding objects have equal mass and the collision is elastic. This situation is nearly the case with colliding billiard balls, and precisely the case with some subatomic particle collisions. We can thus get a mental ima... |
atomic particles, as we shall see in Medical Applications of Nuclear Physics and Particle Physics. Ernest Rutherford, for example, discovered the nature of the atomic nucleus from such experiments. 8.7 Introduction to Rocket Propulsion Learning Objectives By the end of this section, you will be able to: • State Newton’... |
pushes on the ground. If we consider thrust; that is, the force exerted on the rocket by the exhaust gases, then a rocket’s thrust is greater in outer space than in the atmosphere or on the launch pad. In fact, gases are easier to expel into a vacuum. By calculating the change in momentum for the entire system over Δ,... |
Factors Affecting a Rocket’s Acceleration • The greater the exhaust velocity e of the gases relative to the rocket, the greater the acceleration. • The faster the rocket burns its fuel, the greater its acceleration. • The smaller the rocket’s mass (all other factors being the same), the greater the acceleration. Examp... |
s. Solving for 0 / r gives Thus, the mass of the rocket is ln 0 r = e = 11.2×103 m/s 2.5×103 m/s = 4.48 0 r = 4.48 = 88. r = 0 88. (8.103) (8.104) (8.105) This result means that only 1 / 88 of the mass is left when the fuel is burnt, and 87 / 88 of the initial mass was fuel. Expressed as percentages, 98.9% of the rocke... |
orations: Lunar Lander Can you avoid the boulder field and land safely, just before your fuel runs out, as Neil Armstrong did in 1969? Our version of this classic video game accurately simulates the real motion of the lunar lander with the correct mass, thrust, fuel consumption rate, and lunar gravity. The real lunar l... |
not constant over a period of time. Δp = FnetΔ. 8.3 Conservation of Momentum • The conservation of momentum principle is written or ptot = constant ptot = p′tot (isolated system), ptot is the initial total momentum and p′tot is the total momentum some time later. • An isolated system is defined to be one for which the... |
law of motion states that to every action, there is an equal and opposite reaction. • Acceleration of a rocket is = e −. Δ Δ • A rocket’s acceleration depends on three main factors. They are 1. The greater the exhaust velocity of the gases, the greater the acceleration. 2. The faster the rocket burns its fuel, the gre... |
Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion. 13. Can objects in a system have momentum while the momentum of the system is zero? Explain your answer. 14. Must the total energy of a system be conserved whenever its momentum is conserved? Explain why ... |
the intact shell? 21. Professional Application During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain t... |
momentum as the truck? 5. A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the time required for a force of 1500 N to bring the car to rest. 6. The mass of Earth is 5.972×1024 kg and its orbital radius is an average of 1.496×1011 m. Calculate its linear momentum. 8.2... |
m/s and the car plus driver have a mass of 200 kg. You may neglect friction between the car and floor. 12. Professional Application One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater ... |
50×10–2 s. 17. Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the magnitude of the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero. 18. A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a ... |
together. What is their final velocity? 25. Professional Application Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this qu... |
Repeat parts (a) and (b) for the situation in which the ball and the player are going in opposite directions. Might the loss of kinetic energy be related to how much it hurts to catch the pass? 34. A battleship that is 6.00×107 kg and is originally at rest fires a 1100-kg artillery shell horizontally with a velocity o... |
probes may be separated from their launchers by exploding bolts. (They bolt away from one another.) Suppose a 4800-kg satellite uses this method to separate from the 1500-kg remains of its launcher, and that 5000 J of kinetic energy is supplied to the two parts. What are their subsequent velocities using the frame of ... |
×10 – 25 kg (note that the ratio of the masses is 4 to 235). (a) Calculate the velocities of the two nuclei, assuming the plutonium nucleus is originally at rest. (b) How much kinetic energy does each nucleus carry away? Note that the data given here are accurate to three digits only. 40. Professional Application The M... |
ice skates allow her to recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the barbell? (b) How much kinetic energy is gained by this maneuver? (c) Where does the kinetic energy come from? 8.6 Collisions of Point Masses in... |
helium and gold nuclei were 6.68×10−27 kg and 3.29×10−25 kg, respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of 120º during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the go... |
Results Squids have been reported to jump from the ocean and travel 30.0 m (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of 20.0º, assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by... |
less, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away? 8.7 Introduction to Rocket Propulsion 53. Professional Application Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time avai... |
He throws a snowball in the +x-direction, and it travels on a ballistic trajectory, hitting the ground some distance away. Which of the following is true about the boy while he is in the act of throwing the snowball? a. He feels an upward force to compensate for the downward trajectory of the snowball. b. He feels a b... |
change if the figure were falling to a padded or carpeted surface? Explain. 7. A 2.5-kg block slides across a frictionless table toward a horizontal spring.As the block bounces off the spring, a probe measures the velocity of the block (initially negative, moving away from the probe) over time as follows: Table 8.2 Ve... |
the wall for 0.0055 s, and it rebounds from the wall with a speed of 14 m/s in the opposite direction.What is the magnitude of the average force exerted by the wall on the puck? a. 0.308 N b. 0.616 N c. 56 N d. 112 N 12. A 22-g puck hits the wall of an air hockey table perpendicular to the wall with an initial speed o... |
0.50 kg, use the graph to calculate the mass of cart B. Explicitly indicate the principles used in your calculations. d. The students are now asked to Consider the kinetic energy changes in an inelastic collision, specifically whether the initial values of one of the physical quantities affect the fraction of mechanic... |
wall, rebounding off the wall and falling to the ground. The boy is at rest after the collision. What is the momentum of the boy before and after the collision? Is momentum conserved in this collision? Explain. Which of these is an example of an open system and which is an example of a closed system? Explain your answ... |
elastic collision, calculate the change in momentum of the two-car system. (b) If the two cars have a completely inelastic collision, calculate the change in momentum of the two-car system. 23. Puck A (200 g) slides across a frictionless surface to collide with puck B (800 g), initially at rest. The velocity of each p... |
g the collision. c. Energy was lost due to friction between the ball and the floor. d. Energy was lost due to the work done by gravity during the motion. 28. A tennis ball strikes a wall with an initial speed of 15 m/s. The ball bounces off the wall but rebounds with slightly less speed (14 m/s) after the collision. Ex... |
. 3.0 m/s in the –x-direction 354 Chapter 8 | Linear Momentum and Collisions d. 5.0 m/s in the –x-direction 32. Two objects (A and B) of equal mass collide elastically. Mass A is initially moving 4.0 m/s in the +x-direction prior to the collision. Mass B is initially moving 8.0 m/s in the –xdirection prior to the colli... |
velocity of 5 m/s in the –x-direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision? c. Neither momentum nor kinetic energy is conserved. d. More information is needed in order to determine which a. Both will move 0.5 m/s in the –x-direction. b. Mass ... |
the collision is elastic or inelastic. 39. Mass A (1.0 kg) slides across a frictionless surface with a velocity of 8 m/s in the positive direction. Mass B (3.0 kg) is initially at rest. The two objects collide and stick together. What will be the change in the center-of-mass velocity of the system as a result of the c... |
eventually slides to a stop. Which of the following statements is true about this motion? a. Momentum is conserved during the collision, but it is not conserved during the motion before and after the collision. b. Momentum is not conserved at any time during this analysis. c. Momentum is conserved at all times during ... |
s in the –x-direction 50. Mass A is initially moving with a velocity of 15 m/s in the +x-direction. Mass B is twice as massive and is initially moving with a velocity of 10 m/s in the –x-direction. The two objects collide, and after the collision, mass A moves with a speed of 15 m/s in the –x-direction. (a) What is the... |
9.2. The Second Condition for Equilibrium 9.3. Stability 9.4. Applications of Statics, Including Problem-Solving Strategies 9.5. Simple Machines 9.6. Forces and Torques in Muscles and Joints Connection for AP® Courses What might desks, bridges, buildings, trees, and mountains have in common? What do these objects have... |
able to: • State the first condition of equilibrium. • Explain static equilibrium. • Explain dynamic equilibrium. The first condition necessary to achieve equilibrium is the one already mentioned: the net external force on the system must be zero. Expressed as an equation, this is simply net F = 0 Note that if net is ... |
same forces are applied at other points and the stick rotates—in fact, it experiences an accelerated rotation. Here net = 0 but the system is not at equilibrium. Hence, the net = 0 is a necessary—but not sufficient—condition for achieving equilibrium. PhET Explorations: Torque Investigate how torque causes an object t... |
to the door—we push in this direction almost instinctively. Figure 9.7 Torque is the turning or twisting effectiveness of a force, illustrated here for door rotation on its hinges (as viewed from overhead). Torque has both magnitude and direction. (a) Counterclockwise torque is produced by this force, which means that... |
. (a) The three factors,, and for pivot point A on a body are shown here— is the distance from the chosen pivot point to the point where the force is applied, and is the angle between F and the vector directed from the point of application to the pivot point. If the object can rotate around point A, it will rotate coun... |
rotate clockwise, which means the torque for the force shown is clockwise relative to B. Also, the magnitude of the torque is greater when the lever arm is longer. Making Connections: Pivoting Block A solid block of length d is pinned to a wall on its right end. Three forces act on the block as shown below: FA, FB, an... |
(9.6) where net means total. Torques, which are in opposite directions are assigned opposite signs. A common convention is to call counterclockwise (ccw) torques positive and clockwise (cw) torques negative. When two children balance a seesaw as shown in Figure 9.10, they satisfy the two conditions for equilibrium. Mo... |
9.10) Note that a minus sign has been inserted into the second equation because this torque is clockwise and is therefore negative by convention. Since p acts directly on the pivot point, the distance p is zero. A force acting on the pivot cannot cause a rotation, just as pushing directly on the hinges of a door will n... |
location of the seesaw's actual pivot! Several aspects of the preceding example have broad implications. First, the choice of the pivot as the point around which torques are calculated simplified the problem. Since p is exerted on the pivot point, its lever arm is zero. Hence, the torque exerted by the supporting forc... |
F.1.4 The student is able to design an experiment and analyze data testing a question about torques in a balanced rigid system. (S.P. 4.1, 4.2, 5.1) • 3.F.1.5 The student is able to calculate torques on a two-dimensional system in static equilibrium, by examining a representation or model (such as a diagram or physical... |
13 If the pencil is displaced slightly to the side (counterclockwise), it is no longer in equilibrium. Its weight produces a clockwise torque that returns the pencil to its equilibrium position. 366 Chapter 9 | Statics and Torque Figure 9.14 If the pencil is displaced too far, the torque caused by its weight changes di... |
. A cane, a crutch, or a walker increases the stability of the user, even more as the base of support widens. Usually, the cg of a female is lower (closer to the ground) than a male. Young children have their center of gravity between their shoulders, which increases the challenge of learning to walk. Figure 9.18 (a) T... |
earthquakes, and other forces that displace them from equilibrium. Although the examples in this section emphasize gravitational forces, the basic conditions for equilibrium are the same for all types of forces. The net external force must be zero, and the net torque must also be zero. Take-Home Experiment Stand strai... |
the system is in static equilibrium. This condition is always the case when the acceleration of the system is zero and accelerated rotation does not occur. 2. It is particularly important to draw a free body diagram for the system of interest. Carefully label all forces, and note their relative magnitudes, directions,... |
pole with its cg halfway between his hands is shown. Each hand exerts a force equal to half the weight of the pole, = = / 2. (b) The pole vaulter moves the pole to his left, and the forces that the hands exert are no longer equal. See Figure 9.20. If the pole is held with its cg to the left of the person, then he must... |
body diagram for the pole, the system of interest. There is not enough information to use the first condition for equilibrium (net = 0 ), since two of the three forces are unknown and the hand forces cannot be assumed to be equal in this case. There is enough information to use the second condition for equilibrium (ne... |
or a train. Stand facing sideways. How do you move your body to readjust the distribution of your mass as the bus accelerates and decelerates? Now stand facing forward. How do you move your body to readjust the distribution of your mass as the bus accelerates and decelerates? Why is it easier and safer to stand facing... |
). MA = o i (9.29) 380 Chapter 9 | Statics and Torque Figure 9.31 This figure shows that large forces are exerted by the back muscles and experienced in the vertebrae when a person lifts with their back, since these muscles have small effective perpendicular lever arms. The data shown here are analyzed in the preceding... |
equilibrium: from equilibrium a system, when displaced, experiences a net force or torque in the same direction as the displacement Section Summary 9.1 The First Condition for Equilibrium • Statics is the study of forces in equilibrium. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter... |
• Simple machines are devices that can be used to multiply or augment a force that we apply – often at the expense of a distance through which we have to apply the force. • The ratio of output to input forces for any simple machine is called its mechanical advantage • A few simple machines are the lever, nail puller, ... |
to be directly above the person's neck vertebrae. 9.5 Simple Machines 9. Scissors are like a double-lever system. Which of the simple machines in Figure 9.24 and Figure 9.25 is most analogous to scissors? 10. Suppose you pull a nail at a constant rate using a nail puller as shown in Figure 9.24. Is the nail puller in ... |
distance of 0.850m from the hinges. What torque are you exerting relative to the hinges? (b) Does it matter if you push at the same height as the hinges? 2. When tightening a bolt, you push perpendicularly on a wrench with a force of 165 N at a distance of 0.140 m from the center of the bolt. (a) How much torque are y... |
at one end with a force F1 and the other hand holding it up at 50 cm from the end of the plank with force F2. If the plank has a mass of 20 kg and its center of gravity is at the middle of the plank, what are the magnitudes of the forces F1 and F2? 10. A 17.0-m-high and 11.0-m-long wall under construction and its brac... |
magnitude of the force exerted by the hinges on the bridge. 14. A sandwich board advertising sign is constructed as shown in Figure 9.36. The sign's mass is 8.00 kg. (a) Calculate the tension in the chain assuming no friction between the legs and the sidewalk. (b) What force is exerted by each side on the hinge? Figur... |
a force 45 cm from the pivot and the nail is 1.8 cm on the other side? What minimum force must you exert to apply a force of 1250 N to the nail? 20. Suppose you needed to raise a 250-kg mower a distance of 6.0 cm above the ground to change a tire. If you had a 2.0-m long lever, where would you place the fulcrum if you... |
total force do they exert? Figure 9.40 A mass is connected by pulleys and wires to the ankle in this exercise device. 30. A person working at a drafting board may hold her head as shown in Figure 9.41, requiring muscle action to support the head. The three major acting forces are shown. Calculate the direction and mag... |
(a) Using the information in the figure, calculate the force exerted by the lower teeth on the bullet. (b) Calculate the force on the joint. Figure 9.42 The center of mass of the head lies in front of its major point of support, requiring muscle action to hold the head erect. A simplified lever system is shown. 33. A ... |
0.240 m? (d) What is her useful power output if she does 25 pushups in one minute? Figure 9.46 A woman doing pushups. 38. You have just planted a sturdy 2-m-tall palm tree in your front lawn for your mother's birthday. Your brother kicks a 500 g ball, which hits the top of the tree at a speed of 5 m/s and stays in con... |
go-round increases from rest to a constant rotational speed. c. A pendulum swings back and forth. d. A bowling ball rolls down a bowling alley. 2. Five forces of equal magnitude, labeled A–E, are applied to the object shown below. If the object is anchored at point P, which force will provide the greatest torque? Figur... |
experimentally found. Your explanation must include the concept of torque and all steps should be provided in an orderly sequence. You may include a labeled diagram of your setup to help in your description. Include enough detail so that another student could carry out your procedure. 9.5 Simple Machines 9. As a young... |
of raw destructive power. Tornadoes blow houses away as if they were made of paper and have been known to pierce tree trunks with pieces of straw. They descend from clouds in funnel-like shapes that spin violently, particularly at the bottom where they are most narrow, producing winds as high as 500 km/h. (credit: Dap... |
energy, and power are associated with rotational motion. This supports Big Idea 3, that interactions are described by forces. The ability of forces to cause torques (Enduring Understanding 3.F) is extended to the interactions between objects that result in nonzero net torque. This nonzero net torque in turn causes cha... |
to an axis. Extended Knowledge 4.D.2. The angular momentum of a system may change due to interactions with other objects or systems. Extended Knowledge 4.D.3. The change in angular momentum is given by the product of the average torque and the time interval during which the torque is exerted. Big Idea 5. Changes that ... |
angle : = Δ Δ, (10.1) where is the angle of rotation as seen in Figure 10.3. The relationship between angular velocity and linear velocity was also defined in Rotation Angle and Angular Velocity as = (10.2) or =, where is the radius of curvature, also seen in Figure 10.3. According to the sign convention, the counter ... |
1 min 60 sec Entering this quantity into the expression for, we get = Δ Δ = 26.2 rad/s 5.00 s = 5.24 rad/s2. (10.6) (10.7) Strategy for (b) In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for Δ, y... |
motion experiences centripetal acceleration, as seen in Figure 10.5. Thus, t and c are perpendicular and independent of one another. Tangential acceleration t is directly related to the angular acceleration and is linked to an increase or decrease in the velocity, but not its direction. Figure 10.5 Centripetal acceler... |
/s2. (10.15) (10.16) Discussion Units of radians are dimensionless and appear in any relationship between angular and linear quantities. So far, we have defined three rotational quantities—,, and. These quantities are analogous to the translational quantities,, and. Table 10.1 displays rotational quantities, the analog... |
be able to: • Observe the kinematics of rotational motion. • Derive rotational kinematic equations. • Evaluate problem solving strategies for rotational kinematics. Just by using our intuition, we can begin to see how rotational quantities like,, and are related to one another. For example, if a motorcycle wheel has a... |
ational Motion and Angular Momentum Table 10.2 Rotational Kinematic Equations Rotational Translational ¯ = = 0 + = = 0 + (constant, ) = constant, ) 2 + 2 (constant, ) In these equations, the subscript 0 denotes initial values ( 0, 0, and 0 are initial values), and the average angular velocity - and average velocity - a... |
solving problems in linear kinematics. In particular, known values are identified and a relationship is then sought that can be used to solve for the unknown. Solution for (a) Here and are given and needs to be determined. The most straightforward equation to use is = 0 + because the unknown is already on one side and... |
m, about right for when the big fish bites. Figure 10.8 Fishing line coming off a rotating reel moves linearly. Example 10.3 and Example 10.4 consider relationships between rotational and linear quantities associated with a fishing reel. Example 10.4 Calculating the Duration When the Fishing Reel Slows Down and Stops ... |
The distance is very easily found from the relationship between distance and rotation angle: Solving this equation for yields =. = Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities: = (200 rev)2π rad 1... |
number of revolutions, and then the linear distance traveled. = because - is given to be 6.0 rpm. can be used to find Solution ¯ Entering known values into = gives = = 6.0 rpm (2.0 min) = 12 rev. As always, it is necessary to convert revolutions to radians before calculating a linear quantity like from an angular quan... |
when the variables are treated as being clockwise or counterclockwise with respect to a well-defined axis of rotation, and refine the research question based on the examination of data. (S.P. 3.2, 4.1, 5.1, 5.3) • 5.E.2.1 The student is able to describe or calculate the angular momentum and rotational inertia of a sys... |
, =. or = 2 = 2α. (10.40) (10.41) (10.42) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 10 | Rotational Motion and Angular Momentum 401 This last equation is the rotational analog of Newton's second law ( = ), where torque is analogous to force, angular acceleration is analogous to ... |
The general relationship among torque, moment of inertia, and angular acceleration is or net τ = = net τ, (10.43) (10.44) where net is the total torque from all forces relative to a chosen axis. For simplicity, we will only consider torques exerted by forces in the plane of the rotation. Such torques are either positi... |
circle rotate when you added putty at the number 3 (clockwise or counterclockwise)? In which of these directions was the resulting angular velocity? Was the angular velocity constant? What can we say about the direction (clockwise or counterclockwise) of the angular acceleration? How could you change the placement of ... |
force is perpendicular to the radius and friction is negligible, so that =. (10.45) τ = sin θ = (1.50 m)(250 N) = 375 N ⋅ m. Solution for (a) The moment of inertia of a solid disk about this axis is given in Figure 10.12 to be 1 22, where = 50.0 kg and = 1.50 m, so that Now, after we substitute the known values, we fi... |
on it. In terms of revolutions per second, these angular velocities are 2.12 rev/s and 1.41 rev/s, respectively. The father would end up running at about 50 km/h in the first case. Summer Olympics, here he comes! Confirmation of these numbers is left as an exercise for the reader. Making Connections: Multiple Forces o... |
on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors. 10.4 Rotational Kinetic Energy: Work and Energy Revisited ... |
the force times the arc length traveled: net = (net )Δ. (10.53) To get torque and other rotational quantities into the equation, we multiply and divide the right-hand side of the equation by, and gather terms: 406 Chapter 10 | Rotational Motion and Angular Momentum We recognize that net = net τ and Δ / =, so that net ... |
://cnx.org/content/col11844/1.13 Chapter 10 | Rotational Motion and Angular Momentum 407 Figure 10.16 Experimental vehicles, such as this bus, have been constructed in which rotational kinetic energy is stored in a large flywheel. When the bus goes down a hill, its transmission converts its gravitational potential ener... |
its outer edge. Solution for (b) To find from the given information requires more than one step. We start with the kinematic relationship in the equation 2 = 0 2 + 2. (10.64) 408 Chapter 10 | Rotational Motion and Angular Momentum Note that 0 = 0 because we start from rest. Taking the square root of the resulting equa... |
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