Ch.10
5.49 s
A hoop is released from rest at the top of a plane inclined at 21° above horizontal. How long does it take the hoop to roll 26.5 m down the plane?
ΔL/Δt=Fl/2
A metal rod of mass m and length l is pivoted at its center point. Assume the pivot is frictionless. You are free to apply a force F in any manner you choose. What is the maximum rate of change of the rod's angular momentum that you could induce by applying F?
Into the page
A meter stick is hanging from a nail in the wall in your garage, as shown in Figure 10.34. The stick is pulled to the right and released. What is the direction of its angular momentum as it begins to swing down?
64N
A 2- m -long plank is hinged on one end and held in position by a cable attached to the other end, as shown in (Figure 2). If the mass of the plank is 13 kg , what is the tension in the cable?
1600J, 200W −1600 J, −400 W.
An electric motor exerts a constant torque of τ=10 N ⋅ m on a grindstone mounted on its shaft; the moment of inertia of the grindstone is I=2.0 kg ⋅ m2. If the system starts from rest, find the work done by the motor in 8.0 s and the kinetic energy at the end of this time. What was the average power delivered by the motor? Suppose we now turn off the motor and apply a brake to bring the grindstone from its greatest angular speed of 40 rad/s to a stop. If the brake generates a constant torque of 20 N ⋅ m, find the total work done and the average power (including signs) needed to stop the grindstone. Answers:
both the net force acting on the object is zero and the net torque acting on the object is zero
An extended object is in static equilibrium if __________.
The change in the angular displacement over the change in time is the angular velocity.
How is the angular displacement θ related to the angular velocity ω?
The mass of the object. The axis of rotation. The rate at which that the object rotates. The shape of the object.
On what does the angular momentum of an object depend?
the axis of symmetry
The center of gravity of a right circular cylinder or cone lies on ________.
precession
This combined motion of the axis and the flywheel is known as
the positive sense of rotation.
counterclockwise rotation
3.33 s
A particular motor can provide a maximum of 110.0 N ⋅ m of torque. Assuming that all of this torque is used to accelerate a solid, uniform flywheel of mass 10.0 kg and radius 3.00 m, how long will it take for the flywheel to accelerate from rest to 8.13 rad/s?
her angular momentum remains constant.
A spinning ice skater is able to control the rate at which she rotates by pulling in her arms. We can best understand this effect by observing that in this process:
geometric center.
The center of gravity of a homogeneous sphere, cube, circular disk, or rectangular plate is at its
t2>t3>t1
Three forces with equal magnitudes act on the ends of a long, thin rod of length L, as shown in Figure 10.5. The rod is pinned so that it can rotate about its center of mass. Which choice correctly ranks the magnitudes of the torques produced by all three forces, from largest to smallest?
2830 N
A 4-m long, 150-kg steel beam is attached to a wall with one end connected to a hinge that allows the beam to rotate up and down. The other end of the beam is held in a horizontal position with a cable that makes a 27° angle with the beam and is attached to the wall. A mass of 75 kg is hung from the beam 3 meters away from the hinge (see (Figure 2)). Now what is the tension force that keeps this beam in static equilibrium?
30°
A 95 N force exerted at the end of a 0.32 m long torque wrench gives rise to a torque of 15 N ⋅ m. What is the angle (assumed to be less than 90°) between the wrench handle and the direction of the applied force?
The orientation of the force. The magnitude of the force. The distance between the point of force application and the axis of rotation of the object.
On what does the magnitude of an applied torque depend?
The rate of rotation increases by a factor of 2.
For a rotating object experiencing no net external torque, what happens to the rate of rotation if the moment of inertia of the object decreases by a factor of 2?
680N*m
In our first example, we will see that we can use a lever arm to greatly increase the torque without having to provide more force. Suppose, for instance, that an amateur plumber, unable to loosen a pipe fitting, slips a piece of scrap pipe (sometimes called a "cheater") over the handle of his wrench. He then applies his full weight of 900 N to the end of the cheater by standing on it. The distance from the center of the fitting to the point where the weight acts is 0.80 m, and the wrench handle and cheater make an angle of 19°19° with the horizontal (Figure 10.6a). Find the magnitude and direction of the torque of his weight about the center of the pipe fitting.
acm, y= 2/3g T=1/3Mg acm, y= g/2 T=Mg/2
In our first moving-axis example we will consider a yo-yo (like the one we analyzed in Example 9.9). As shown in Figure 10.12a, the yo-yo consists of a solid disk with radius R and mass M. In our earlier analysis, we used energy considerations to find the yo-yo's speed after it had dropped a certain distance. Now let's find the acceleration of the yo-yo and the tension in the string. Note that these quantities cannot be calculated using the conservation-of-energy analysis alone. If the solid cylinder is replaced by a thin cylindrical shell with the same mass and radius as before, find acm,y and T. Answers:
1/8T
Let's consider again the whirling block shown in Figure 10.21. By letting additional string pass through the hole in the plane, we increase the radius to twice its original value. If the initial tension in the string is T, then the tension in the string after the radius is increased is
6.2kg m/s2 0.96J Answer: 0.62J
Now let's apply our definition of angular momentum to a specific object. A mobile sculpture is suspended from the ceiling of an airport terminal building. It consists of two metal spheres, each with mass 2.0 kg, connected by a uniform metal rod with mass 3.0 kg and length s=4.0 m. The assembly is suspended at its midpoint by a wire and rotates in a horizontal plane, making 3.0 revolutions per minute. Find the angular momentum and kinetic energy of the assembly. What would the kinetic energy of the sculpture be if the rod was twice as long but it still had the same angular momentum as in the example?
0.30 rad/s.
Suppose the bullet goes through the door and emerges with a speed of 100 m/s. Find the final angular velocity of the door just after the bullet goes through it. Answer:
at the same rate.
Suppose we replace both hover pucks with pucks that are the same size as the originals but twice as massive. Otherwise, we keep the experiment the same. Compared to the pucks in the video, this pair of pucks will rotate
90 cm (10 cm from the weight)
Suppose we replace the mass in the video with one that is four times heavier. How far from the free end must we place the pivot to keep the meter stick in balance?
ω = 1.3 rev/s
The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. (See the figure below (Figure 1).) When the skater's hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thin-walled hollow cylinder. His hands and arms have a combined mass of 8.0 kg. When outstretched, they span 1.8 m; when wrapped, they form a cylinder of radius 25 cm. The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.35 kg⋅m2 . If the skater's original angular speed is 0.45 rev/s , what is his final angular speed?
a1>a2>a3
Three cylinders, each with a total mass of M and an outer radius of R, are shown in Figure 10.9. In cylinder 1, the mass is uniformly distributed. Cylinders 2 and 3 are identical hollow cylinders, with an inner radius of R/2. All three cylinders can rotate about their centers of mass. A force with a magnitude of F acts on cylinders 1 and 2 at the outer radius and on cylinder 3 at its inner radius. Which choice correctly ranks the magnitudes of the angular accelerations of the three cylinders, from smallest to largest?
20 N⋅m
To exercise, a man attaches a 4.0 kg weight to the heel of his foot. When his leg is stretched out before him, what is the torque exerted by the weight about his knee, 50 cm away from the weight? Use g = 10 m/s^2.
108N 260N
To strengthen his arm and chest muscles, a 75-kg athlete 2.0 m tall is doing a series of push-ups as shown in the figure below (Figure 1). His center of mass is 1.20 m from the bottom of his feet, and the centers of his palms are 30.0 cm from the top of his head. Find the force that the floor exerts on each of his feet, assuming that both feet exert the same force. Find the force that the floor exerts on each of his hands, assuming both palms exert the same force.
2m to the right of the pivot
Two children want to balance horizontally on a seesaw. The first child is sitting one meter to the left of the pivot point located at the center of mass of the seesaw. The second child has one-half the mass of the first child. Where should the second child sit to balance the seesaw?
The force applied at the doorknob exerts the greater torque.
Two equal forces are applied perpendicular to a door. The first force is applied at the midpoint of the door; the second force is applied at the doorknob. Which force exerts the greater torque?
look at pic!
Which of the following objects is in static equilibrium?
C=D, A,B, E C,D
You are using a wrench to rotate a bolt around its center. Consider all the forces in the figure below, indicated by the arrows, to have the same magnitude. Rank the scenarios in terms of the magnitude of torque applied, from smallest to largest torque. Which of the following scenarios as illustrated in Part B apply no torque?
Angular momentum
the analog of the linear momentum (p→=mV) of a particle.
x =0.28 m
A 5.0 kg plank is being held up in a horizontal position by a rope as shown in (Figure 1). The plank is 1.0 m long, and its left end is attached to the wall by a hinge. A 10 kg box is placed on the plank at a distance x from the hinge. If the maximum tension that the rope can sustain is 60 N , what is the maximum value of x before the rope breaks?
The sphere.
A disk and a sphere are released simultaneously at the top of an inclined plane. They roll down without slipping. Which will reach the bottom first?
1620 N
A 4-m long, 150-kg steel beam is attached to a wall with one end connected to a hinge that allows the beam to rotate up and down. The other end of the beam is held in a horizontal position with a cable that makes a 27° angle with the beam and is attached to the wall (see (Figure 1)). What is the tension force that keeps this beam in static equilibrium?
h′ = 2.0m
A bucket with mass m = 1.0 kg is suspended over a well by a winch and rope (Figure 1). The winch consists of a solid cylinder with mass 4.0 kg and radius R = 0.10 m about which the rope is wrapped. A handle is attached to one end in order to rotate the cylinder. For the purposes of this example, we are going to ignore any frictional forces in the winch. Now suppose that the winch handle breaks off, allowing the bucket to fall to the water as the rope unwinds from the cylinder. Assume that the bucket is released at t=0 and the water level is at a depth h = 2.4 m below the bucket at t=0 . How far above the water is it at 0.50 s ?
7.6 N ⋅ m
A force of 17 N is applied to the end of a 0.63 m long torque wrench at an angle 45° from a line joining the pivot point to the handle. What is the magnitude of the torque generated about the pivot point?
The angular speed of the system increases. The moment of inertia of the system decreases.
A girl moves quickly to the center of a spinning merry-go-round, traveling along the radius of the merry-go-round. Which of the following statements are true?
The side the boy is sitting on will tilt downward.
A heavy boy and a lightweight girl are balanced on a massless seesaw. The boy moves backward, increasing his distance from the pivot point. What happens to the seesaw?
Yes The tension in the cord is directed toward the axis of rotation and produces no torque. There is no net external torque. ω = 3.64 rad/s ΔK = 3.16×10−3J W =3.16×10−3J
A small block on a frictionless horizontal surface has a mass of 0.0280 kg. It is attached to a massless cord passing through a hole in the surface. (See the figure below (Figure 1).) The block is originally revolving at a distance of 0.265 m from the hole with an angular speed of 1.50 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.170 m. You may treat the block as a particle. Is angular momentum conserved? How do you know? What is the new angular speed? Find the change in kinetic energy of the block. How much work was done in pulling the cord?
40 kg ⋅ m2/s
A uniform, solid flywheel of radius 1.4 m and mass 15 kg rotates at 2.7 rad/s. What is the magnitude of the flywheel's angular momentum?
a = 0.5gsinθ
A very thin circular hoop of mass m and radius r rolls without slipping down a ramp inclined at an angle θ with the horizontal, as shown in the figure. (Figure 1)
5.7 m
An irregularly shaped object 10 m long is placed with each end on a scale. If the scale on the right reads 96 N and the scale on the left reads 71 N, how far from the left is the center of gravity.
W = 1885 J Krot,f = 1885 J ωf = 8.68 rad/s Δt = 17.4 s Pavg = 109 W P=2∗Pavg
Consider a motor that exerts a constant torque of 25.0 N⋅m to a horizontal platform whose moment of inertia is 50.0 kg⋅m^2 . Assume that the platform is initially at rest and the torque is applied for 12.0 rotations . Neglect friction. How much work W does the motor do on the platform during this process? What is the rotational kinetic energy of the platform Krot,f at the end of the process described above? What is the angular velocity ωf of the platform at the end of this process? How long Δt does it take for the motor to do the work done on the platform calculated in Part A? What is the average power Pavg delivered by the motor in the situation above? Note that the instantaneous power P delivered by the motor is directly proportional to ω , so P increases as the platform spins faster and faster. How does the instantaneous power Pf being delivered by the motor at the time tf compare to the average power Pavg calculated in Part E?
They have the same kinetic energy.
Cylinders A and B have the same radius but different masses, mA= 2mB. A constant torque τ is applied to each cylinder. If they start from rest, which cylinder has the largest kinetic energy after they have each rotated through 10 revolutions?
0.36m/s2 Answer: 7.5 N.
Figure 10.10a shows the same situation that we analyzed in Example 9.7. A cable is wrapped several times around a uniform solid cylinder with diameter 0.12 m and mass 50 kg that can rotate freely about its axis. The cable is pulled by a force with magnitude 9.0 N. Assuming that the cable unwinds without stretching or slipping, find the magnitude of its acceleration. To wind the winch back up, you exert a force that points straight down on its rightmost edge. What is the magnitude of the force you need to exert if you wind the winch with an angular acceleration of 5.0 rad/s2?
280rad/s =2700 rev/min
Figure 10.33 shows a top view of a gyroscope wheel in the form of a solid cylinder, driven by an electric motor. The pivot is at O, and the masses of the axle and motor are negligible. Is the precession clockwise or counterclockwise, as seen from above? If the gyroscope takes 4.0 s for one revolution of precession, what is the wheel's angular velocity?
5.5 degrees
If the pipe fitting under consideration can withstand a maximum torque of 700 N ⋅ m700 N ⋅ m without breaking, what is the maximum angle at which our amateur plumber can safely stand on the end of the cheater and apply his full weight without breaking the pipe fitting?
Right hand rule
If you curl the fingers of your right hand in the direction the object rotates, then your thumb points in the direction of the vector angular velocity ω→ and angular momentum L→. In the case of a torque, you curl the fingers of your right hand in the direction the torque would turn the body if acting on its own; then your thumb points in the torque's direction.
x=1.2m Answer: 1.24 m from the left end.
Let's begin by determining the equilibrium position of a seesaw pivot. You and a friend play on a seesaw. Your mass is 90 kg, and your friend's mass is 60 kg. The seesaw board is 3.0 m long and has negligible mass. Where should the pivot be placed so that the seesaw will balance when you sit on the left end and your friend sits on the right end? If the board has a mass of 20 kg, where should the pivot be placed for balance?
Question19
Look &see
250N -200N 44N magnitude force at elbow= 200N Answer: 510 N.
Now let's consider a biological example of static equilibrium. Figure 10.28a shows a human arm lifting a dumbbell. The forearm is in equilibrium under the action of the weight w of the dumbbell, the tension T in the tendon connected to the biceps muscle at point A, and the forces exerted on the forearm by the upper arm at the elbow joint. For clarity, the tendon force has been displaced away from the elbow farther than its actual position. The weight w and the angle θ are given. Find the tension T in the tendon and the two components of force (Ex and Ey) at the elbow (three unknown quantities in all). Ignore the weight of the forearm itself. Evaluate your results for w=50 N, d=0.10 m, l=0.50 m, and θ=80°. Double the weight of the dumbbell. What is the new tension T in the tendon connected to the biceps muscle at point B? Answer: 510 N.
v= √=2gh/1+M/2m Answers: (a) m=M/2, (b) T=mg/2=Mg/4.
Now let's go back to the old-fashioned well in Example 9.8. In that example we were limited to a conservation-of-energy analysis of the system, which gave us only the speeds of the bucket and the winch at certain points in the motion. Now we can use Newton's second law for rotation to find the acceleration of the bucket (mass m) and the angular acceleration of the winch cylinder. a) What is the relationship between m and M if the bucket's acceleration is half the acceleration of free fall? (b) In this case, what is the rope tension T?
fs=2/7MgsinB Answers: acm, x=2/3 g sin β; μs=1/3 tan β.
Now we will look at one of the most important examples of rotation about a moving axis. A bowling ball rolls without slipping down the return ramp at the side of the alley (Figure 10.13a). The ramp is inclined at an angle β to the horizontal. What is the ball's acceleration? What is the friction force acting on the ball? Treat the ball as a uniform solid sphere, ignoring the finger holes. Suppose we replace the bowling ball with a solid cylinder. Determine the acceleration of the cylinder and the minimum coefficient of static friction if the cylinder rolls down the ramp without slipping. Answers:
tall narrow glass, tapered glass, short, fat glass
Rank in order, from least stable to most stable, the three glasses of water shown.
The light person's end will go down.
Suppose that a heavy person and a light person are balanced on a teeter-totter made of a plank of wood. Each person now moves in toward the fulcrum a distance of 25 cm. What effect will this have on the balance of the teeter-totter?
make it twice as massive
The artist Anya Calderona constructs the mobile shown in the figure.(Figure 1) In the illustrated configuration, the mobile is perfectly balanced. Assume the strings and crossbars are massless. If Anya decides to make the star twice as massive, and not change the length of any crossbar or the location of any object, what does she have to do with the mass of the smiley face to keep the mobile in perfect balance? Note that she may have to change masses of other objects to keep the entire structure balanced.
W = 67.6 N θ = 84.1∘ Fmax = 135N θ = 0∘
The device shown in the figure below (Figure 1) is one version of a Russell traction apparatus. It has two functions: to support the injured leg horizontally and at the same time provide a horizontal traction force on it. This can be done by adjusting the weight W and the angle θ. For this patient, his leg (including his foot) is 80.0 cm long (measured from the hip joint) and has a mass of 13.6 kg. Its center of mass is 34.0 cm from the hip joint. A support strap is attached to the patient's ankle 13.0 cm from the bottom of his foot. What weight W is needed to support the leg horizontally? If the therapist specifies that the traction force must be 14.0 N horizontally, what must be the angle θ ? What is the greatest traction force that this apparatus could supply to this patient's leg? What is θ in that case?
moment arm (or lever arm)
The distance l1 plays the role of the wrench handle, and we call it the _________ of force F→1 about O.
He spins slower.
The experimenter from the video rotates on his stool, this time holding his empty hands in his lap. You stand on a desk above him and drop a long, heavy bean bag straight down so that it lands across his lap, in his hands. What happens?
torque
The quantitative measure of the tendency of a force to cause or change a body's rotational motion is called
D>B=E>F>A>C
The wrench in the figure has six forces of equal magnitude acting on it. (Figure 1) Rank these forces (A through F) on the basis of the magnitude of the torque they apply to the wrench, measured about an axis centered on the bolt.
angular velocity= 0.40 rad/s initial energy= 800J Final energy= 0.40J KE not conserved.
This example demonstrates how a particle moving in a straight line can carry angular momentum. Specifically, we will see that the straight-line motion of a particle can, in fact, induce rotational motion after an impact. A uniform door 1.0 m wide with a mass of 15 kg is hinged at one side so that it can rotate without friction about a vertical axis. The door is unlatched. A police detective fires a bullet with a mass of 10 g and a speed of 400 m/s into the exact center of the door in a direction perpendicular to the plane of the door. Find the angular velocity of the door just after the bullet is embedded in it. Is kinetic energy conserved?
None of the blocks are in equilibrium.
Three rectangular blocks rest on a horizontal sheet of ice. The blocks are free to both slide along the ice and rotate about their centers of mass as they slide. Figure 10.25 shows how one or more forces, all with a magnitude of F, act along the short end of each block. Which block is in equilibrium?
If there is no net torque acting on it.
Under what condition is the angular momentum of an object conserved?
negative.
counterclockwise torques are positive and clockwise torques are
the total torque on a body due to its weight is the same as though all the weight were concentrated at the center of mass of the body.
in a uniform gravitational field,
constant. (Conservation of angular momentum)
the total angular momentum of an isolated system is
920 N
A 4-m long, 150-kg steel beam is attached to a wall with one end connected to a hinge that allows the beam to rotate up and down. The other end of the beam is held in a horizontal position with a cable that makes a 27° angle with the beam and is attached to the wall. A mass of 75 kg is hung from the beam 3 meters away from the hinge (see (Figure 2)). What is the vertical component of the force that the hinge exerts on the beam?
The object will rotate with constant angular acceleration.
A constant net torque is applied to a rotating object. Which of the following best describes the object's motion?
a.) n1=268 N, fs=268N b.) 0.27 c.) 1020N, 75 degrees Answers: 1160 N, 58°.
In this example, we will consider the classic static-equilibrium situation of a ladder leaning against a wall. Suppose a medieval knight is climbing a uniform ladder that is 5.0 m long and weighs 180 N. The knight, who weighs 800 N, stops a third of the way up the ladder (Figure 10.27a). The bottom of the ladder rests on a horizontal stone ledge and leans across the castle's moat in equilibrium against a vertical wall that is frictionless because of a thick layer of moss. The ladder makes an angle of 53° with the horizontal, conveniently forming a 3-4-5 right triangle. (a) Find the normal and friction forces on the ladder at its base. (b) Find the minimum coefficient of static friction needed to prevent slipping. (c) Find the magnitude and direction of the contact force on the ladder at the base. What are the magnitude and direction of the contact force on the ladder at the base if the knight has climbed two-thirds of the way up the ladder?
a = 0.527 m/s^2 a = 0.527 m/s^2 α = 5.27 rad/s^2 TB = 20.7N TA =41.7N
The figure below (Figure 1) illustrates an Atwood's machine. Let the masses of blocks A and B be 4.50 kg and 2.00 kg, respectively, the moment of inertia of the wheel about its axis be 0.400 kg⋅m2 and the radius of the wheel be 0.100 m. Find the linear acceleration of block A if there is no slipping between the cord and the surface of the wheel. Find the linear acceleration of block B if there is no slipping between the cord and the surface of the wheel. Find the angular acceleration of the wheel C if there is no slipping between the cord and the surface of the wheel. Find the tension in left side of the cord if there is no slipping between the cord and the surface of the wheel. Find the tension in right side of the cord if there is no slipping between the cord and the surface of the wheel.
zero,
The torques due to all the internal forces add to
τA = 0 τB = +Fbcosθ τC = −Fcsinθ τD = +Fdsin(ϕ−θ)
A force F of magnitude F making an angle θ with the x axis is applied to a particle located along axis of rotation A, at Cartesian coordinates (0,0) in the figure. The vector F⃗ lies in the xy plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the xy plane. (Figure 1) What is the torque τA about axis A due to the force F⃗ ? What is the torque τB about axis B due to the force F⃗ ? (B is the point at Cartesian coordinates (0,b) , located a distance b from the origin along the y axis.) What is the torque τC about axis C due to F⃗ ? (C is the point at Cartesian coordinates (c,0) , a distance c along the x axis.) What is the torque τD about axis D due to F⃗ ? (D is the point located at a distance d from the origin and making an angle ϕ with the x axis.)
ω = 8.00 rad/s I =21.1 kg⋅m2
A light rope is wrapped several times around a large wheel with a radius of 0.375 m. The wheel rotates on frictionless bearings about a stationary horizontal axis, as shown in the figure (Figure 1). The free end of the rope is tied to a suitcase with a mass of 19.5 kg. The suitcase is released from rest at a height of 4.00 m above the ground. The suitcase has a speed of 3.00 m/s when it reaches the ground. Calculate the angular velocity of the wheel when the suitcase reaches the ground. Calculate the moment of inertia of the wheel.
1.92×10^−2 kg⋅m2/s 7.68*10^-2 kg*m^2/s
A majorette in a parade is performing some acrobatic twirlings of her baton. Assume that the baton is a uniform rod of mass 0.120 kg and length 80.0 cm . Initially, the baton is spinning about an axis through its center at angular velocity 3.00 rad/s . (Figure 1)What is the magnitude of its angular momentum about a point where the axis of rotation intersects the center of the baton? With a skillful move, the majorette changes the rotation of her baton so that now it is spinning about an axis passing through its end at the same angular velocity 3.00 rad/s as before. (Figure 2) What is the new magnitude of the angular momentum of the rod about a point where the axis of rotation intersects the end of the baton?
PLcosθ.
A person pushes vertically downward with force P⃗ on a lever of length L that is inclined at an angle θ above the horizontal as shown in the figure below (Figure 1). The torque that the person's push produces about point A is
4.2 kg
A potter's wheel, with rotational inertia 24 kg ⋅ m2, is spinning freely at 40.0 rpm. The potter drops a lump of clay onto the wheel, where it sticks a distance 1.2 m from the rotational axis. If the subsequent angular speed of the wheel and clay is 32 rpm, what is the mass of the clay?
L= 0 L= 55 kg m/s
A small, 4.5-kg brick is released from rest 3.00 m above a horizontal seesaw on a fulcrum at its center, as shown in the figure below (Figure 1). Find the angular momentum of this brick about a horizontal axis through the fulcrum and perpendicular to the plane of the figure the instant the brick is released. Find the angular momentum of this brick about a horizontal axis through the fulcrum and perpendicular to the plane of the figure the instant before it strikes the seesaw.
2vi
In Figure 10.21, a block slides in a circular path on a horizontal frictionless plane under the action of a string that passes through a hole in the plane and is held vertically underneath it. By pulling on the lower end of the string, we reduce the radius to half of its original value. If the initial speed of the block is υi,υi, then the speed of the block after the string is shortened is
initial= 64J Final= 320J
In this example, we see how a system can have constant angular momentum without having a constant angular velocity! A physics professor stands at the center of a turntable, holding his arms extended horizontally, with a 5.0 kg dumbbell in each hand (Figure 10.22). He is set rotating about a vertical axis, making one revolution in 2.0 s. His moment of inertia (without the dumbbells) is 3.0 kg ⋅ m2 when his arms are outstretched, and drops to 2.2 kg ⋅ m2 when his arms are pulled in close to his chest. The dumbbells are 1.0 m from the axis initially and 0.20 m from it at the end. Find the professor's new angular velocity if he pulls the dumbbells close to his chest, and compare the final total kinetic energy with the initial value.
3,2,1,4
There are four ice skaters. Three skaters are spinning with equal rates of rotation, while the fourth skater is not spinning at all. Rank the skaters in terms of their angular momentum.