Physics Chapter 8 Rotational Motion
A person grips the end of the wrench, 0.5 meters from the bolt, and pushes on the end in the direction of the bolt with a force of 50 N. What is the torque exerted on the bolt?
0 N × m. The torque would be 25 N m if the force were perpendicular to the wrench. Since the force is in the same direction as the wrench, there is no component of the force perpendicular to the wrench, so the torque is zero.
Axis
1) straight line about which rotation takes place, 2) straight lines for reference in a graph, usually the x axis for measuring horizontal displacement and the y axis for measuring vertical displacement.
A boy and a girl are playing on a teeter totter. The boy is three times as massive as the girl. If the boy is sitting 0.5 meters away from the pivot point, how far away should the girl sit such that the two can sit balanced, horizontally?
1.5 m The torque goes as weight times distance, so the distance of the girl from the pivot point must be three times higher than that of the boy.
A person grips a wrench 0.2 meters away from the bolt and pulls in a direction perpendicular to the wrench with a force of 50 N. What is the torque exerted on the bolt?
10 N m Since the force is perpendicular to the wrench, the torque is given by torque = force times distance.
Which will have the greater acceleration rolling down an incline: a hoop or a solid disk?
A solid disk will. Both will rotate about the central axis and the shape that has the most of its mass far from its axis is the hoop. Therefore, it has greater rotational inertia and is harder to start rolling.
mechanical equilibrium
A state in which an object is not accelerating. Where ∑t=0.
Three people are standing on a spinning merry-go-round. Person A is standing near the center, person B is standing halfway between the center and the edge, and person C is standing near the edge. Which person takes the longest amount of time to complete one circle?
All three complete the circle in the same amount of time. This means that person C is moving the fastest since he covers a larger circumference than the others.
What is the law of inertia for rotating systems in terms of angular momentum?
An object in rotational motion will maintain the same angular momentum unless it is acted upon by a torque.
Distinguish between linear momentum and angular momentum.
Angular momentum depends on the distribution of mass, whereas linear momentum depends on the total mass.
Centrifugal force.
Apparent outward force on a rotating or revolving body
As distance increases between most of the mass of an object and its center of rotation, does rotational inertia increase or decrease?
As distance increases between most of the mass of an object from the center, rotational inertia increases. The greater the rotational inertia the greater the difficulty in changing its rotational state. For example a barbell is harder to rotate versus a dumbbell.
How can gravity be simulated in an orbiting space station?
By either rotating the space station or accelerating the space station linearly. The rotation method is sustainable for longer and it gives people in the station the illusion that gravity is pulling them towards the outside of the space station.
centripetal force
Center-directed force that causes an object to follow a curved or circular path.
An ice skater is spinning while holding her arms out perpendicular to her body. As she brings her arms towards her body, the rotational speed
Due to angular momentum being conserved, the rotational speed needs to increase if the moment of inertia decreases.
centripetal force formula
F = mv²/r
What is the relationship between the center of gravity and the support base for an object that is in stable equilibrium?
For an object in stable equilibrium the support base is placed at the center of gravity in order to balance the torques applied to the object by gravity.
If a skater who is spinning pulls her arms in so as to reduce her rotational inertia by half, by how much will her angular momentum change? By how much will her rate of spin increase? (Why do your answers differ?)
Her angular momentum will remain the same because of the conservation of momentum. Momentum before = momentum after. The rate of spin, RPM, will be double because she halved the rotational inertia and doubled her rotational velocity.
Rotational inertia of simple pendulum.
I = mr²
If you hang at rest by your hands from a vertical rope, where is your center of gravity with respect to the rope?
If you hang from a rope by your hands your center of gravity will be right under the rope (and about half way down your body).
equilibrium
In general, a state of balance. For mechanical equilibrium, the state in which no net forces and no net torques act. In liquids, the state in which evaporation equals condensation. More generally, the state in which no net change of energy occurs.
Is it an inward force or an outward force that is exerted on the clothes during the spin cycle of an automatic washing machine?
Inward force. The clothes have some velocity and they have a tendency to continue moving in whatever direction they are presently moving. This means a force inward needs to be applied to keep them from flying out while the washing machine is spinning.
How does the lever arm change when you decrease the distance to the nut?
It decreases
How does the lever arm change if you decrease the angle of the force?
It decreases.
Is it easier for a circus performer to balance a long rod held vertically with people hanging off the other end, or the same long rod without the people at the other end, and why?
It is easier for the performer to balance a long rod held vertically with people at the other end because the rotational inertia is greater.
Consider balancing a hammer upright on the tip of your finger. The head is likely heavier than the handle. Is it easier to balance with the end of the handle on your fingertip, with the head at the top, or the other way around?
It is easier to stand the handle on your finger tips and the head at the top. This way there is greater rotational inertia and it will be more resistant to rotational change.
Is it easier to balance a long rod with a mass attached to it when the mass is closer to your hand or when the mass is farther away?
It is easier when the mass is farther from your hand.
If you toss a stick into the air, it appears to wobble all over the place. Specifically, about what place does it wobble?
It wobbles about it's center of mass
Which is easier to get swinging: a baseball bat held at the narrow end or a bat held closer to the massive end (choked up)?
Its easier to get a bat to swing if you hold it closer to the massive end. This is because there is greater distance at the narrow end which provides greater rotational inertia.
What is rotational inertia, and how is it similar to inertia as studied in earlier chapters?
Just as discussed in earlier chapters, rotational inertia means that an object rotating about an axis tends to remain rotating about the same axis unless interfered with by some external influence.
Where is the center of mass of Earth's crust?
Like a giant basketball, Earth's crust is a spherical shell with its center of mass at Earth's center.
tangent
Line that touches a curve in one place only and is parallel to the curve at that point.
RPM
Means revolutions per minute.
How is a flywheel constructed to maximize its rotational inertia?
Most of the mass is concentrated far from the axis.
If you are not wearing a seat belt in a car that rounds a curve, and you slide across your seat and slam against a car door, what kind of force is responsible for your slide: centripetal, centrifugal, or no force? Why is the correct answer "no force"?
No force causes you to slam into your car door. It is natural to continue moving in a straight line unless you are acted upon by a force.
If the string that holds a whirling can in its circular path breaks, what kind of force causes it to move in a straight-line path: centripetal, centrifugal, or no force? What law of physics supports your answer?
No force. The can is already in motion, by the law of inertia, it will keep moving until acted upon by an external force.
Can an object have more than one center of gravity?
No. A rigid object has one CG. If an object is nonrigid, such as a piece of clay or putty, and is distorted into different shapes, then its CG may change as its shape changes. Even then, it has one CG for any given shape
linear speed
Path distance moved per unit of time. Also just called speed.
lever arm
Perpendicular distance between an axis and the line of action of a force that tends to produce rotation about that axis.
Fulcrum
Pivot point of a lever.
center of mass
Point at the center of an object's mass distribution, where all its mass can be considered to be concentrated. For everyday conditions, it is the same as the center of gravity.
center of gravity
Point at the center of the object's weight distribution, where the force of gravity can be considered to act.
Angular momentum
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 is the product of mass, speed, and radial distance of rotation. Angular momentum = mass × velocity × radial distance.
torque
Product of force and lever-arm distance, which tends to produce rotational acceleration. Torque - lever-arm distance × force
linear momentum
Product of the mass and the velocity of an object. Also called momentum. This definition applies at speeds much less than the speed of light.
Rotational inertia
Reluctance or apparent resistance of an object to change its state of rotation, determined by the distribution of the mass of the object and the location of the axis of rotation or revolution.
Inertia depends on mass; rotational inertia depends on mass and something else. What?
Rotational inertia depends on the distribution of the mass about the axis of rotation. The greater the distance between an object's mass concentration and the axis the greater the rotational inertia.
Why does the rotational inertia of the rod with the attached mass closer to your hand compare the way it does with the rotational inertial of the rod with the attached mass farther away?
Rotational inertia depends on whether the mass is farther or closer to the point of rotation. The farther the mass is, the higher the rotational inertia.
On a rotating turntable, does tangential speed or rotational speed vary with distance from the center?
Rotational speed does not vary with distance. Rotational speeds is the measurement of revolutions per unit of time. Tangential speed depends on radial distance. It is measured as: Tangential speed ~ radial distance × rotational speed. v~rw
Rotational velocity
Rotational speed together with a direction for the axis of rotation or revolution.
Consider the balanced seesaw in Figure 8.18. Suppose the girl on the left suddenly gains 50 N, such as by being handed a bag of apples. Where should she sit in order to be in balance, assuming the heavier boy does not move?
She should sit ½ m closer to the center. Then her lever arm is 2.5 m. This checks: 300 N × 2.5 m = 500 N × 1.5 m.
Imagine a ladybug sitting halfway between the rotational axis and the outer edge of the turntable in Figure 8.1b. When the turntable has a rotational speed of 20 RPM and the bug has a tangential speed of 2 cm/s, what will be the rotational and tangential speeds of her friend who sits at the outer edge?
Since all parts of the turntable have the same rotational speed, her friend also rotates at 20 RPM. Tangential speed ~ radial distance × rotational speed. Since the friend is twice as far from the axis of rotation, she moves twice as fast—4 cm/s.
Rotation
Spinning motion that occurs when an object rotates about an axis located within the object, usually an axis through its center of mass.
stable equilibrium
State of an object balanced so that any small displacement or rotation raises its center of gravity.
linear motion
Straight-line motion, as opposed to circular, angular, or rotational motion.
What are the units of measurement for tangential speed? For rotational speed?
Tangential speeds is measured in meters per second. mwh/s. Rational speed is measured as the number of rotations per unit of time. Typically expressed as revolutions per minute or RPM.
Why doesn't the Leaning Tower of Pisa topple over?
The Leaning Tower of Pisa does not topple over because its center of mass is over its base. If its center of mass was not above its base then it would tip over, unless the ground somehow provided a counter-torque.
Two blocks are attached on either side of a pulley. Block A has a mass of m and Block B has a mass of 2m and is lower than Block A. Which of the following statements correctly describes the system shown in the figure?
The angular acceleration of the pulley is nonzero.
In terms of center of gravity, support base, and torque, why can't you stand with your heels and back to a wall and then bend over to touch your toes and return to your stand-up position?
The center of gravity moves when you move forward so the center of gravity isn't in line with the support base. So when you're against a wall you tip over.
Where is the center of mass of a baseball? Where is its center of gravity? Where are these centers for a baseball bat?
The center of mass and center of gravity for a baseball are in the center - the mass is distributed uniformly in spherical objects. The center of mass and gravity for a baseball bat is on the heavier side.
Where is the center of mass of a hollow soccer ball?
The center of mass of a soccer ball is in the center of the ball (in the core). This is because a soccer ball is spherically symmetric.
Two blocks are attached on either side of a pulley. Block A has a mass of m and Block B has a mass of 2m and is lower than Block A. What happens when block B moves downward.
The cord's tension on the right side of the pulley is higher than on the left side. If the pulley were stationary (as in many systems where only linear motion is studied), then the tension in the cord on both sides of the pulley would be equal; however, if the pulley rotates with a certain angular acceleration, as in the present situation, the tensions must be different. If they were equal, the pulley could not have an angular acceleration.
Why is it dangerous to slide open the top drawers of a fully loaded file cabinet that is not secured to the floor?
The file cabinet is in danger of tipping because the center of gravity may extend beyond the support base. If it does, then torque due to gravity tips the cabinet.
The diameter of the base of tapered cup is 6 cm. The diameter of the mouth is 9 cm. The path of cup curves when you roll it. Which end, the base or the mouth, rolls faster. How much faster?
The greater the radius or diameter, the greater the tangential speed. So the wide part rolls faster. 9/6 = 3/2 = 1.5 times faster
What is meant by the "lever arm" of a torque?
The lever arm about any axis of rotation is the perpendicular distance from the axis to the line along which the force acts. Therefore, it is the shortest distance between the applied force and the rotational axis. Torque is defined as the product of the lever arm and the force that produces rotation: Torque = lever arm × force.
A uniform meterstick supported at the 25-cm mark balances when a 1-kg rock is suspended at the 0-cm end. What is the mass of the meterstick?
The mass of the meterstick is 1 kg.
How does the rotational inertia of the rod with the mass toward the bottom compare with the rotational inertia of the mass toward the top?
The rotational inertia of the rod with the mass closer to the top is greater than the rotational inertia of the rod with the mass closer to the bottom.
When Dr. Hewitt cuts the broom right through the center of gravity, how do the weights of the two sides of the broom compare?
The shorter side, where the bristles of the broom are, has a greater weight than the handle. The longer side (the handle) has a greater weight than the bristle side.
A record player is spinning with a certain rotational speed. An ant is standing somewhere on the record without slipping. Suddenly, the rotational speed of the record player is doubled (it is spinning twice as quickly). What happens to the ant's tangential speed?
The tangential speed doubles. The ant travels around the same circle in half the time it used to take, so the tangential speed is twice as high
How does the tapered rim of a wheel on a railroad train allow one part of the rim to have a greater tangential speed than another part when it is rolling on a track?
The tapered rim means that some points of the wheel have a greater tangential speed. The wide part provides a greater speed while the narrow part has less. So, when a train rounds a curve, wheels on the outer track ride the wide part of the tapered rims, while the opposite wheels ride on the narrow parts. This way wheels have different tangential speeds for the same rotational speed. Otherwise, scraping would occur.
How does the torque due to the weight of one side of the broom exerted around the balance point compare with the torque exerted by the weight of the other side of the broom around the balance point?
The torque due to the weight of the shorter side (the bristles of the broom) is equal in magnitude to the torque due to the weight of the longer side, and opposite in direction.
Trains ride on a pair of tracks. For straight-line motion, both tracks are the same length. Not so for tracks along a curve, however. Which track is longer: the one on the outside of the curve or the one on the inside?
The track on the outside of the curve is longer—just as the circumference of a circle of greater radius is longer.
A tapered cup rolled on a flat surface makes a circular path. What does this tell you about the tangential speed of the rim of the wide end of the cup compared with that of the rim of the narrow end?
The wide part of the cup rolls faster than the narrow part, because it has greater tangential speed than the narrow end, so it rolls in a curve.
If you are not wearing a seat belt in a car that rounds a curve, and you slide across your seat and slam against a car door, what kind of force is responsible for your slide: centripetal, centrifugal, or no force?
There is no force as viewed by someone outside the car. To them you move in a straight line.
If a pipe effectively extends a wrench handle to three times its length, by how much will the torque increase for the same applied force?
Three times more leverage for the same force produces three times more torque. (Caution: This method of increasing torque sometimes results in shearing off the bolt!)
What does a torque tend to do to an object?
Torques tend to twist or change the rotational motion of things. If you want a stationary object to rotate or a rotating object to change rotational velocity, apply torque.
Inertial frame of reference
Unaccelerated vantage point in which Newton's laws hold exactly
Precession
Wavering of a spinning object, such that its axis of rotation traces out a cone.
How do clockwise and counterclockwise torques compare when a system is balanced?
When a system is balanced it is said to be in equilibrium. This means the clockwise and counterclockwise torques are equal and therefore cancel one another out.
When a car drives off a cliff, why does it rotate forward as it falls?
When all the wheels are on the ground, the car's CG is above a support base and no tipping occurs. But when the car drives off a cliff, the front wheels are first to leave the ground and the car is supported only by the rear wheels. The CG then extends beyond the support base, and rotation occurs. Interestingly, the speed of the car is related to how much time the CG is not supported and, hence, the amount the car rotates while it falls.
conservation of angular momentum
When no external torque acts on an object or a system of objects, no change of angular momentum takes place. hence, the angular momentum before an event involving only internal torques is equal to the angular momentum after the event.
Why does bending your legs when running enable you to swing your legs to and fro more rapidly?
When you bend your legs you reduce their rotational inertia so that they can rotate back and forth more quickly. Long-legged people tend to walk with slower strides than those with short legs. Giraffes, horses, and ostriches run with a slower gait than dachshunds, mice and bugs.
When you whirl a can at the end of a string in a circular path, what is the direction of the force you exert on the can?
When you whirl a can in circular motion you apply a force inward. By applying a force in, along the string, you are continually changing the direction of the velocity of the can. This means you provide a centripetal acceleration, which points directly into the center of the can.
CG
abbreviation from center of gravity
The center of gravity of a basketball is located _______.
at its geometrical center
Where is the center of gravity of the broom that Dr. Hewitt holds up?
at the balance point. The balance point is at the center of the heavier side of the broom.
Why is centrifugal force in a rotating frame called a "fictitious force"?
because it is just the lack of a force to keep an object in rotational motion.
The torque exerted by a crowbar on an object increases with increased _______.
force and leverage distance
The tangential speed on the outer edge of a rotating carousel is _______.
greater than toward the center
Where is the center of gravity of a donut?
in the hole
How does the force needed to turn the wrench change if you increase the lever arm?
it decreases
angular momentum units
kgm²/s
Torque units
m × N
centripetal force equation
mass * tangential speed² / radius.
Tangential Speed Formula
radial distance * rotational speed symbolized as: rω
rotational speed units
revolutions / min. rev/seconds. degree/min. radian/sec.
angular momentum formula
rotational inertia * angular velocity
A record player is spinning with a certain rotational speed. One ant is standing half way between the center of the record and the edge, and another ant is standing on the edge. Which ant is moving faster?
the ant on the edge. Since the ant on the edge travels around a bigger circle in the same amount of time as the other ant, it is moving faster.
The moment of inertia of a rotating ice skater is largest when
the arms are held perpendicular to her body
Tangential speed
the linear speed tangent to a curved path, such as in circular motion
