Physics Chapter 8
axis (pl. axes)
(a) Straight line about which rotation takes place. (b) Straight lines for reference in a graph, usually the x-axis for measuring horizontal displacement and the y-axis for measuring vertical displacement.
Re call the equilibrium rule in Chapter 2--that the sum of the forces acting on a body or any system must equal zero for mechanical equilibrium. That is, ∑F = 0. We now see an additional condition. The *net torque* on a body or on a system must also be zero for mechanical equilibrium...
...(∑T = 0, where T stands for torque). Anything in mechanical equilibrium doesn't accelerate--neither linearly nor rotationally.
an unbalanced torque unbalanced forces and torques are usually thought of as the "causes" of accelerations, momenta, speeds
A ball rolls down a hill mainly because of a balanced torque. its angular momentum. its angular acceleration. its rotational inertia. an unbalanced torque.
above the fulcrum Justification: Because the fulcrum is always located below the center of gravity.
A baseball bat is balanced on a fulcrum. The center of gravity of the bat is located -near the heavy end. -above the fulcrum. closer to the handgrip.
-highest, farthest from your hand. Justification: Because with the heavier weight of the broom father away from your hand, this gives the broom more rotational inertia. This prevents the broom from rotating off your finger as easily, because it has more of the property to resist change.
A broom is easier to balance on its end when the heavier end (the brush end) is -nearest your hand. -highest, farthest from your hand. -same either way
directed toward the center of the curve. Justification: Any force directed toward a fixed center is *centripetal force* and in the case of this car, as long as the car in a circle with constant speed like bucket on a string around their head then the n there is a force constantly pulling the car inward toward the center, just like the string of the bucket.
A car travels in a circle with constant speed. The net force on the car is -directed toward the center of the curve. -directed forward, in the direction of travel. -zero because the car is not accelerating. -none of these
coin. Justification: This is because the coins mass is less spread out among itself, opposed to the ring that has all of its mass spread out and concentrated on its outer parts away from its center of gravity. This gives the ring a greater rotational inertia, so it takes the ring while to gain speed because objects that have a large amount of inertia resist change in motion.
A coin and a ring roll down an incline starting at the same time. The one to reach the bottom first will be the -ring. -coin. -Both reach the bottom at the same time.
two times the other's Justification: Since the second flywheel is twice the size of the first then it has twice the amount of mass. Even though it is the same size and shape, mass is the measurement of inertia. This means that the second flywheel has more rotational INERTIA
A flywheel's mass is twice that of another of the same size and shape. The more massive flywheel's rotational inertia is -the same as the other's. -two times the other's. -four times the other's. -half the other's.
rotates faster. Justification: By coming together the particles are lowering the rotational inertia. And according to according to the law of conservation of angular momentum, the lowering of rotational inertia raises rotational velocity.
A huge rotating cloud of particles in space gravitate together to form an increasingly dense ball. As it shrinks in size, the cloud rotates faster. cannot rotate. rotates at the same speed. rotates slower.
sphere Justification: This is because the sphere's mass is the least spread out among itself, opposed to the ring and the disk that have all of their mass spread out and concentrated on its specific parts away from its center of gravity. This gives the sphere a least rotational inertia, so it takes the ring the least time to gain speed because objects that have a large amount of inertia resist change in motion.
A ring, a disk, and a solid sphere begin rolling down a hill together. The one to reach the bottom first is the -sphere. -ring. -disk. -They all reach the bottom at the same time. -Not enough information is given.
None of the above choices would work. Justification: The heavier mother should move closer to HER LIGHTER SON since he's putting less force on the seesaw, and she's putting more she need to lower the length of her torque arm by moving closer to the fulcrum
A small boy places a rock under the middle of a of a long wood plank, sits near one end and his mother sits near the opposite end. To balance each other -both should move closer to the ends of the plank. -the mother should move further away from the boy. -the boy should move closer to his mother. -both should move closer to the middle of the plank. -None of the above choices would work.
rotation Justification: Torque is the rotational counterpart of *force*. So since force changes the motion of an object, Torque is just the name for the amount of force that changes the *rotational motion* of an object.
A torque acting on an object tends to produce -a center of gravity. -rotation. -linear motion. -velocity. -equilibrium.
Just as an external net force is required to change the linear momentum of an object, an external net torque is required to change the angular momentum of an object. We can state a rotational version of Newton's first law (the law of inertia):
An object or system of objects will maintain its angular momentum unless acted upon by an external net torque.
Centrifugal force
An outward force apparent in a rotating frame of reference. It is apparent (fictitious) in the sense that it is not part of an interaction but is a result of rotation--with no reaction-force counterpart.
= rotational inertia × rotational velocity
Angular momentum is defined as the product of rotational inertia and rotational velocity. Like linear momentum, angular momentum is a vector quantity and has direction as well as magnitude. Angular momentum =
Linear momentum = mass × velocity
Angular momentum's counterpart is:
The first is at the central core parallel to the length of the pencil, where the lead is. The second is perpendicular to the midpoint of the pencil; and the third axes is perpendicular to one end of the pencil. Rotational inertia is very small at the first position, and it's easy to rotate the pencil between your fingertips because most of the mass is very close to the axis of the pencil. The second axis, is greater because it's weight is distributed more. The third axis, which is at the end of the pencil swings like a pendulum, rotational inertia is even greater than the second axis.
Consider three axes of rotation for a pencil: along the lead; at right angles to the lead at the middle; at right angles to the lead at one end. Rate the rotational inertias about each axis from small to large.
the resultant forces and torques must both be zero. Justification: For something to be in MEchanical equilibrium then all forces must cancel out in to zero. and torque is just rotational force, another force that must be at zero
For a system in mechanical equilibrium -the resultant forces and torques must both be zero. -the resultant torques must be zero. -the resultant forces and torques must be equal. -the resultant force must be zero.
Angular momentum = mvr (also = Iω)
For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as the magnitude of its linear momentum, *mv*, multiplied by the radial distance, *r*. In shorthand notation:
near the outside Justification: This is because a point farther from the center travels a longer path in the same time and therefore has a greater tangential speed.
Horses that move with the fastest linear speed on a merry-go-round are located -near the outside. -anywhere, because they all move at the same speed. -near the center.
When Clockwise moment is equal to the anti-clockwise moment they have about the same torque, and remain in equilibrium. This is because both torque cancel each other out.
How do clockwise and counterclockwise torques compare when a system is balanced?
double Justification: This is because rotational speed is directly proportional to tangential speed. Although the linear speed of an object moving in a circular path, (which is tangential speed) is relative to its location at each point. Any time the rotational speed changes it effects the tangential speed. Which in this case would be the speed of the hamster
If a turntable's rotational speed is doubled, then the linear speed of a pet hamster sitting on the edge of the record will -double. -halve. -remain the same.
increase Justification: When the Earth is spinning, we experience a centrifugal force that slightly decreases are weight, it slightly pushes us away from earth. if the Earth were to slow down then this force would decrease and we would increase weight.
If the Earth rotated more slowly about its axis, your apparent weight would -stay the same. -increase. -be zero. -decrease.
increase Justification: Angular momentum = rotational inertia X Rotational velocity (according to the equation) This means when rotational inertia goes down, then rotational velocity goes up. So as jupiter's mass becomes less spread out, it rotational inertia goes down, and this in turn raised its rotational velocity.
If the planet Jupiter underwent gravitational collapse, its rate of rotation about its axis would increase. decrease. stay the same. more information needed
bristles part Justification: Justification: This is because Lever arm × Force =Torque. The center of gravity where the fulcrum was located was there to compromise for the extra weight of the bristle and so, it's lowering the lever arm there. like the fat kid on the seesaw. SO if you cut the broom in half there, the bristle end will still be heavier.
If you balance a broom horizontally on one finger, the center of gravity of the broom will be above your finger - closer to the bristles end than the handle end. If you saw the broom in two pieces at that point and weigh the two parts on a scale, you'll find that the heavier part is the -bristles part. -handle part. -both the same weight
number of rotations or revolutions per unit of time
It is common to express rotational rates in revolutions per minute (RPM).
*revolution*
Motion of an object turning around an axis that lies outside the object.
*1/3 the distance from the fulcrum*. Justification: This is because Lever arm × Force =Torque. So since the boy is 3 times heavier he's putting 3 times the force on the lever, which equals to a higher torque according to the equation. To balance the seesaw with the other boy who's putting less 3 times less force on it, he has to sit 1/3 closer to the fulcrum. Because the farther he's from the fulcrum the higher the integer that needs to be entered into the lever arm part of the equation.
On a balanced seesaw, a boy three times as heavy as his partner sits -more than 1/3 the distance from the fulcrum. -1/3 the distance from the fulcrum. -less than 1/3 the distance from the fulcrum.
smaller than that of the wide part Justification: This is because a point closer to the center travels a shorter path in the same time and therefore has a smaller tangential speed.
Since each rolling wheel of a railroad train is tapered, the narrow part of the wheel has a tangential speed that is -greater than that of the wide part. -the same as that of the wide part. -smaller than that of the wide part.
*rotational*
Spinning motion that occurs when an object rotates about an axis located within the object (usually an axis through its center of mass).
Center of mass (CM)
The average position of the mass of an object. The CM moves as if all the external forces acted at this point.
Center of gravity (CG)
The average position of weight or the single point associated with an object where the force of gravity can be considered to act.
the left truck Justification: If a line is drawn straight down from the center of gravity of an object and it falls within the inside of it's base, then it is with in equilibrium. Meaning it will stay balanced in this state.
The centers of gravity of the three trucks parked on a hill are shown by the Xs. Which truck(s) will tip over? -the left truck -the middle truck -the right truck
decreases Justification: This is because as the disk is getting larger it's spreading it mass farther and farther out. This means its gaining more rotational inertia. Therefore the pizza disc beginning to resist change and is decreasing in speed.
The chef at the infamous Fattening Tower of Pizza tosses a spinning disk of uncooked pizza dough into the air. The disk's diameter increases during the flight, while its rotational speed -increases. -remains constant. -decreases.
above a place of support Justification: If a line is drawn straight down from the center of gravity of an object and it falls within the inside of it's base, then it is with in equilibrium. Meaning it will stay balanced in this state. This is what is occurring with the tower of Pisa
The famous Leaning Tower of Pisa doesn't topple over because its center of gravity is -stabilized by its structure. -above a place of support. -in the same place as its center of mass. -relatively low for such a tall building. -displaced from its center.
Tangential speed
The linear speed tangent to a curved path, such as in circular motion.
center of gravity: Justification: A rigid object only has one center of gravity. A non-rigid object such as a monkey that can move also can only have one center of gravity, but depending on the way the monkey changes its position or shape, that center its center of gravity can be repositioned. Like in the case with it's tail.
The long, heavy tail of a spider monkey enables the monkey to easily vary its -inertia. -weight. -center of gravity. -momentum. -none of these
Rotational speed (sometimes called angular speed)
The number of rotations or revolutions per unit of time; often measured in rotations or revolutions per second or per minute. (Scientists usually measure it in radians per second.)
Angular momentum
The product of a body's rotational inertia and rotational velocity about a particular axis. For an object that is small compared with the radial distance, it can be expressed as the product of mass, speed, and the radial distance of rotation.
Torque
The product of force and lever-arm distance, which tends to produce rotation. Torque = lever arm × force
Rotational inertia (often called moment of inertia)
The property of an object that measures its resistance to any change in its state of rotation: if at rest, the body tends to remain at rest; if rotating, it tends to remain rotating and will continue to do so unless acted upon by a external net torque.
Equilibrium
The state of an object in which it is not acted upon by a net force or a net torque.
wide handle Justification: According to Diameter × Force =Torque. Diameter is inversely proportional to force, so the the wider the lever arm, in this case the handle of the screwdriver, the lower the force has to be.
To turn a stubborn screw, it is best to use a screwdriver that has a long handle. wide handle. smooth handle. none of these
center of mass When an object is thrown in the air symmetrical or not, the weight around it always moves around its center of mass. So when a baseball bat is thrown in the air although it look like its wobbling all over the place its center of mass stays around the same place while its following its trajectory.
Toss a baseball bat into the air and it wobbles about its -heavier end. -geometrical center. -center of mass.
when all wheels are on the ground, the car's center of gravity is above a support base. But, when the car drives off a cliff, the front wheels are first to leave the ground and the support base shrinks to the line between the rear wheels. So the car's center of gravity extends beyond the support base and it rotates.
When a car drives off a cliff it rotates forward as it falls. For a higher speed off the cliff, will it rotate more, or less? (Consider the time that the unbalanced torque acts.)
increases Justification: according to the law of conservation of angular momentum, If no external net torque acts on a rotating system, then the angular momentum of the system remains the same. in this case the ice skater changes the net torque of her rotating system by lowering her rotational inertia when she brings her arms in, this raises her rotational velocity .
When a twirling ice skater brings her arms inward, her rotational speed -decreases. -remains the same. -increases.
balled up Justification: Angular momentum = rotational inertia X Rotational velocity (according to the equation) This means when rotational inertia goes down, then rotational velocity goes up. In Other words when you bring your arms and legs into lower rotational velocity, then you will spin faster because your rotational velocity will go up
When doing somersaults, you'll more easily rotate when your body is straight with both arms at your sides. balled up. straight with both arms above your head. no difference
Conservation of angular momentum
When no external torque acts on an object or a system of objects, no change of angular momentum can occur. Hence, the angular momentum before an event involving only internal torques or no torques is equal to the angular momentum after the event.
This is due to Centrifugal force which is not a "real" force. The force that you feel while being pressed against the door is due to your inertia as the car turns. According to Newton's third law an object in motion tends to stay in motion. So your body keeps going in one direction, while the car moves in another pushing the car door against you. The car door provides a centripetal force to make you turn.
When you are in the front passenger seat of a car turning to the left, you may find yourself pressed against the right-side door. Why do you press against the door? Why does the door press on you? Does your explanation involve a centrifugal force, or Newton's laws?
outside horse Justification: This is because a point farther from the center travels a longer path in the same time and therefore has a greater tangential speed.
Which moves faster in m/s on a merry-go-round: a horse on the inside or a horse on the outside near the outer rail? -inside horse -outside horse -Both move at the same speed in m/s.
water Because the molecules in ice are more spread out they will have a higher amount of rotational inertia causing the the can to role slower if it were water because the the H2o molecules are closer together.
Which will roll down an incline in the shortest time, a can filled with water or the same can filled with ice? -ice -water -both the same -Not enough information is given.
Centripetal force
force A force directed toward a fixed point, usually the cause of circular motion : F = mv²/r.
Tangential speed is directly proportional to rotational speed at every fixed distance that from the axis of rotation. For example, on a merry go round, the faster its rotational speed the faster the tangential speed of any person on the ride.
hat is the relationship between tangential speed and distance from the center of the rotational axis? Give an example.
Tangential speed~
~radial distance × rotational speed. In symbol form, v ~ rω
*RPM*¹
¹Physics types usually describe rotational speed, ω, in terms of the number of "radians" turned in a unit of time. There are a little more than 6 radians in a full rotation (2π radians, to be exact). When a direction is assigned to rotational speed, we call it rotational velocity). Rotational velocity is a vector whose magnitude is the rotational speed. By convention, the rotational velocity vector lies along the axis of rotation.
The rotational inertia of a pole, or of any object, depends on the axis about which it rotates.⁴
⁴When the mass of an object is concentrated at the radius r from the axis of rotation (as for a simple pendulum bob or a thing ring), rotational inertia I is equal to the mass m multiplied by the square of the radial distance. For this special case, I = mr².
