rotation & torque

REMINDERS: Solving Torque

(1) Because any point can be chosen as the origin for the position vector, it is best to choose a point that has a force already acting so we can eliminate that force from net torque equation (2) for an object sitting on top of another at one point there will be a contact(normal) force at that point. [Ex: fulcrum point] if the two objects are in contact at many points we can say that the net contact(normal) force would be located at a symmetrical point between the two objects. (3) remember that weight and mg are different forces that act on an object. Weight is actually normal force upwards and mg is force due to gravity.

right-hand rule

(1) Point the fingers of your right hand in the direction of position vector r (2) Curl your fingers toward the force vector F. (if you are able to curl your fingers natural towards vector F, your thumb points in the direction of torque, which should he out of the plane of the page which corresponds to positive value of torque or counterclockwise rotation) (if you are unable to naturally curl your fingers towards vector F, flip your hand 180 degrees and point in direction of position vector r. You should now be able to curl your fingers natural. negative torque, clockwise rotation)

Two interesting facts about center of gravity

(1) an object will be balanced(meaning the object will not rotate or tip over) if it is supported at the center of gravity or any point on the vertical line above or below it; (2) if an irregularly shaped object is hung or supported from any position (or point), the objects center of gravity will always lie somewhere on the straight line above or below the position or point of support.

How is artificial gravity created in a rotating space station?

(1) artificial gravity is created by the normal force of the space station on the person. The normal force prevent the person from flying off tangential. Normal force points towards the center of the station. To the person this force seems centrifugal pushing him down to the wall( or ground) but the normal force is correcting for his tangential velocity and forcing him to follow the circular path of rotation. (2) In artificial gravity the person feels Earth normal force. Since the station walls(or from the person's perspective the station's floor) is rotating at a certain speed so the person standing on the wall moves with a certain tangential speed . The person want's to fly off on a straight line tangent with the velocity given by the rotating station. However, the person is not allowed to fly off and follows the circular path because the wall(floor) is following a circular path.

what are the torque parameters?

(1) point O, the origin, is usually chosen to coincide with the axis of rotation. while this is commonly done it is NOT a requirement. Other points besides the axis of rotation can be chosen. (2) in calculating torque, forces acting IN THE PLANE perpendicular to the axis of rotation are the only forces that contribute to torque; (ex: for a rotating merry-go-round, normal force of the rider, mg of a rider would not contribute to torque calculation because these forces do not affect rotation. (3) when the first two parameters are met, rotation can only be counterclockwise( + torque) or clockwise ( - torque) (4) multiple torques acting on an object are additive. (5) if net torque is not zero, an object experiences a change in rotational speed or an object experiences rotational acceleration (6) if the net torque is zero, object is either not rotating or rotating at a constant rotational speed.

Under what two conditions will lever arm be used?

(1) when the Force acting on the object is not perpendicular to its' position vector (2) when you have multiple points not all along the same straight line (position vector) and no consistent line where all force act. in this scenario lever arm would allow you to solve for multiple torques as if they were all acting upon the same line.

Solving the TILTED AXIS

(1)Draw the freebody diagram with normal axis.(think PLUS sign) (2) Imagine rotating the axis so as to line up with the tilted axis frame of reference (3) redraw angles on new tilted axis.

angular displacement

(1)the difference in an objects final and initial angle measured in radians; (2) the angle measures in radians through which a body rotates in a circular path;

moment of inertia

(1)the mass property of a rigid body that determines the torque needed for a desired angular rotation. (2) moment of inertia is the sum of each mass particle times the square distance to the axis of rotation.

QQ8.7 If global warming continues, it's likely that some ice from the polar ice caps of the Earth will melt and the water will be distributed closer to the Equator. If this occurs, would the length of the day (one revolution) (a) increase, (b) decrease, or (c) remain the same?

(a) Increase.. Since angular momentum is conserved, the rotational speed would decrease due to the redistribution of mass from poles to equator. There is a inversely proportional relationship between rotational speed and Time for a given arc length traveled according. Therefore, the period increase with decreasing rotational speed.

When comparing a smaller gear to a larger gear: (a) does a smaller gear increase or decrease torque? why? (b) does a smaller gear increase or decrease bike speed? why?

(a) Torque decrease because the radius is now smaller. Torque is defined as radius times Force. If the applied force is the same but you decrease the radius, the value of Torque also decreases. (b) Bike speed increases. Lets consider two bikes. Bike A has two gears of equal size. Bike B has one gear that is the same size as bike A and its other gear[the rear gear] is half its size. Lets assume that each bike's gears are connected to each other via chain. Lets also assume that the second gear is fastened to back wheel so that the gear and wheel rotate the same number of times. Said another way if the back gear rotates three times the back wheel also rotates three times. In bike A, for every one rotation of the first gear the second gear & wheel also rotates once. In bike B for every one rotation of the first gear the smaller gear & WHEEL rotates twice. Since the smaller gear is fastened to the axle, the gear and wheel rotate the same number of times. If the time it takes for the rotation of the first gear in both bikes is the same the 2nd bike will travel further during that time period because the wheel[gear] rotates twice. Since speed is the distance traveled over time and the time is constant the 2nd bike moves faster.

Identify pseudonyms for the following terms: (a.) angular velocity (b.) angular acceleration

(a) rotational velocity or velocity of rotation (b) rotational acceleration or acceleration of rotation.

QQ8.5 Which arrives at the bottom first? (a) a ball rolling without sliding down an incline (b) a solid cylinder rolling without sliding down the incline, (c) a box of the same mass as the ball sliding down a frictionless incline having the same dimensions as A? Assume each object is released from rest at the top of its incline.

(c) In option C, the initial mgh is converted to translational KE, while in option A & B, some of the initial mgh is converted into rotational KE and lessens the value of translational at the end. The ball would be second to reach the end because its moment of inertia is less than a solid cylinder resulting in a smaller rotational KE at then end. As a result the translational KE would be slightly larger for the same initial PE.

1 radian equals how many degrees? how do you orientate yourself to the radians from degrees

1 radian approximately equals 57.29 degrees. we know that pi = 180 degrees like 2 pi = 360 degrees. Pi has a value of 3.14. So 3.14 radians =180 degrees. Pi/2(1.57 radians) = 90 degrees; pi/4(0.785 radians) = 45 degrees;

The higher the gear the faster a bike can move. Why is it difficult to start pedaling a bike in a higher gear?

A bike in a higher gear requires more Force because its radius is smaller. Lets consider two bikes in different gears. Let Bike A be in first gear and bike B in third. Gears are fixed to the axle of a wheel so that the gear and wheel rotate at the same rate. Torque causes the gear (and wheel) to rotate. Lets say that 10 units of torque are need to get the gear to spin its fastest. Torque equals force times radius. The larger gear has a bigger radius so less force is needed to achieve the 10 units of torque. The small gear has a smaller radius so more force is required to get to the 10 units of torque.

Explain how force, torque, rotational speed, and angular acceleration relate to a 5-speed bike. Assume that the bike starts from rest.

A constant force is applied to a pedal causing it to rotate. The pedal is fastened to a gear so that the gear and pedal rotate at the same speed. The pedal gear is connected to a rear gear via a chain. In the same way that the first gear is fastened to the pedal so to is the rear gear attached to the back wheel[via the axle]. As a result, the rear gear and rear wheel will also rotate at the same speed. Since the bike was initially at rest the gears[& wheels] all experience an increase in rotational speed until a maximum value is reached that depends on the tangential chain speed. Once this maximum chain speed is reached, the bike will not move any faster on this gear. The chain is then moved to the next gear witch has a smaller radius. This action has two effects. (1) The tangential speed of the chain decreases & (2) the bike becomes more difficult to pedal. The chain speed decreases because of the decrease in gear radius. Tangential speed is the product of the radius & (rotational) angular speed. Both gears have the same rotational speed because they are fixed into the axis. However the second gear is smaller and thus has a smaller radius value, thus a smaller tangential speed. As for the 2nd effect, the bike becomes more difficult to pedal because more Force is required to get the object to rotate. Torque equals the product of Force & radius. A bike requires a certain torque value to achieve the maximum tangential chain speed. Let's consider a hypothetical bike where 10 units of torque are needed to get to the maximum tangential speed. When we shift to the smaller gear the value of torque decreases. [For our hypothetical bike, let's say it decreases to 6 units of torque]. If the rider wants to maintain the 10 units of torque that he was at before shifting gears he must now exert a larger force because he is limited by the fact that the radius of the new gear is smaller. If he were to continue to exert the same force, the bike would slow down. To get back up to the 10 units of torque, the rider exert more force [or experiences more resistance] on the pedal.

(1)How does a massive star collapsing under the influence of gravitational force demonstrate concepts of angular momentum? (2)How does this cause a gigantic outburst of energy supernova)?

Any star whether it is a planet, sun, or moon rotates about its own axis. Just like the earth rotates about itself so to do stars, rotate about their own axis. A star like the sun has energy. This energy is provided by the nuclear fusion reaction of helium and hydrogen. When a star runs out of energy, gravitational forces cause the star to collapse upon itself. Since momentum(specifically angular momentum) is conserved, the star rotates at a faster rate because the mass has moved closer to the origin. when the star collapses upon itself all the particles come into contact and ignite a large nuclear fusion reaction. this fusion reaction results in a large gigantic outburst of energy(supernova).

CQ7 In some motorcycle races, the riders drive over small hills, and the motorcycle becomes airborne for a short time. If the motorcycle racer keeps the throttle open while leaving the hill and going into the air, the motorcycle's nose tends to rise upwards. Why does this happen

As the motorcycle leaves the ground, the friction between the tire and the ground suddenly disappears. if the motorcycle driver keeps the throttle open while leaving the ground, the rear tire will increase its angular speed and, hence its angular momentum. The airborne motorcycle is now an isolated system, and its angular momentum must be conserved. The increase in angular momentum of the tire directed, say clockwise must be compensated for by an increase in angular momentum of the entire motorcycle counterclockwise. This rotation results in the nose of the motorcycle rising and the tail dropping.

Consider opening a revolving door. Where on the door would it be easiest to apply a force to open the door? At the door's hinge or at the door's knob. Explain why this is so.

At the knob. The knob is further away from the axis of rotation and therefore a given force will generate the largest torque (rotating force)

Why does a ball move in a circle?

Because of inertia(Newton's first law) the ball wants to move in a straight line; However by exerting a radial force the string prevents the motion along a straight line and forces the ball to curve and follow the circular path. This tension force in the string changes the velocity vectors direction. If the force where to vanish the object would immediately leave its circular path and move along a straight line tangent to the circle at the point where the force vanished.

why is normal force zero for a roller coaster that just makes it to the top?

Because the object is just about to free fall down to the ground. when an object free falls there is no normal force acting only mg. observation(1) is that if the roller coaster just makes it over the top the velocity of the roller coaster must have slowed down. Observation(2) there must be some negative tangential acceleration in addition to the centripetal acceleration since the object slowed down.

Under what conditions will the center of mass lie on the axis of symmetry?

Bodies the are homogeneous and symmetrical.

what is centripetal force?

Centripetal force is not some new force that acts upon an object. it' is a descriptor of some force that already exist and is acting on the system towards the center (axis of rotation). it forces an object to follow a circular path instead of flying of tangentially Ex the gravitational force acting on a satellite rotating and the tension force in the string of a rotating yo-yo are described as centripetal forces because they are acting on the object keeping it in circular motion and are center seeking in nature.

QQ 8.1 Using a screwdriver, you try to remove a screw from a piece of furniture but can't get it to turn. To increase the chances of success you should use a screwdriver that (a) is longer (b) is shorter (c) has a narrower handle or (d) has a wider handle.

Correct answer is (d). A wider handle is equivalent to increasing your radius (distance from the axis of rotation). Since torque is defined as radius times Force, the same Force with a wider handle generates a large torque, or larger rotational Force. .The more torque you have the easier it is to turn.

What is true about a rigid object rotating about a fixed axis?

Every portion of the object has the same angular speed and same angular acceleration; however the tangential speed and acceleration take different values depending on the distance from the given point of rotation

True or False Torques values must be calculated from the specific axis of rotation.

False. torques can be calculated from any point. In fact torques can even be calculated even if the object is not rotating at all.

[ANSWER] How can point R [pivot point on a wall] have an x & y component?

Firstly, lets establish that there is a pin in the beam at the wall, when considering rotational motion, the beam can rotate at two different locations. If the rope was not attached to the right end of the beam, the beam could rotate around the pin(left end). The pin would be the origin of this rotation. If the left end of the beam has its pin removed, the beam would rotate around the right end (rope-beam attachment point) . The right end (rope-beam attachment point) would be the origin of rotation. The point R has an x-component because the pin exerts a force leftward to counteract the rope rightward tension force pushing the beam to the right. The pin exerts an upward force R to counteract the force that wants the left side of the beam to fall down( or rotate clockwise)

For a given Torque, what happens when the radius or force is changed? What happens to torque if the Force is constant but the radius changes?

For a given torque(rotational force) if you increase the radius(distance from the axis of rotation) less force is needed to produce the torque. If you increase the radius and keep the Force the same, the amount of torque increases.

REVISITING PAST CONCEPTS how should we think to determine the direction of static friction?

For an object not moving one should think if there were no static friction, which direction would the object move. Then you should know that the Force of static friction acts in the direction opposite For an object moving as the result of static friction, think the static friction points in the direction due to inertia (I think)

centripetal acceleration

For circular motion (at constant speed) the acceleration vector that points towards the center of the circle.

What's the difference between center of mass and center of gravity?

If the body(object) is so large that gravity varies significantly over the body, then the center of gravity will be different than the center of mass. Except for this scenario, there is no appreciable difference between center of gravity and center of mass.

There is no such thing as centrifugal force Ex :Ball on a String (explanation)

If you have ever done this yourself, you know that you feel a force pulling outward on your hand. This is not what's happening. To keep the ball moving in a circle you pull inwardly on the string, which in turn exerts the force on the ball. The ball exerts an equal and opposite force and this is the force your hand feels. If centrifugal force existed the ball would fly outwardly. But it doesn't the ball flies off tangentially in the direction of the velocity it had at the moment it was released.

CQ1 Why can't you put your heels firmly against a wall and then bend over without falling?

In order for you to not fall, your center of gravity must be on the same line as your support point, your feet. If your heals are against the wall, when you bend over your center of gravity is no longer on the same line(directly above) your support point. As a result you lose your balance.

What condition(s) must be true for an object to be in equilibrium?

Net force in x & y direction must equal zero and net torque must also equal zero

[ANSWER] Is the FBD drawn correctly for the pulley in part (a)?

No it is not. The FBD are drawn with respect to the each object. If we were to draw the FBD for the pulley, the tension forces would point in the opposite direction as currently represented.

Consider the following passage: "When a diver or an acrobat wishes to make several somersaults, she pulls her hands and feet close to the trunk of her body in order to rotate at a greater angular speed. IN THIS CASE, THE EXTERNAL FORCE DUE TO GRAVITY ACTS THROUGH HER CENTER OF GRAVITY AND HENCE EXERTS NO TORQUE ABOUT HER AXIS OF ROTATION, SO THE ANGULAR MOMENTUM ABOUT HER CENTER OF GRAVITY IS CONSERVED. For Example, when a diver wishes to double her angular speed, she must reduce her moment of inertia to half its initial value." Explained the capitalized statement. How does the diver differ from the figure skater? Explain how no torque is acting on the skater and net torque is zero for the diver because gravity acts through her center of gravity.

No torque is exerted because the force due to gravity on the diver is acting at the origin and thus has a radius of zero. So the force due to gravity to at 0 distance from axis of rotation is zero torque... The diver differs from the skater in that the diver does experience a net force that could affect rotation, gravitational force. However, this force does not affect rotation because the force acts at the center of rotation which means net torque for the system is zero. The skater is already spinning and has no torque acting on her to increase or decrease her rotation. So net torque is zero and the law of conservation of angular momentum still applies. She did exert a force initially to get her to spin, but that force is no longer acting at the point where she varies her arms to change her rotational speed.

Is the gravity we experience on the face of the earth a centripetal force?

No. gravity is not acting centripetally for us because while gravity is acting it is not acting to keep us traveling in a circular path around the earth. Nor are we traveling with some velocity independent of being in contact with the earth's rotation. With that said gravity would be acting centripetally for a satellite in orbit around the earth. The satellite is traveling with some velocity(independent of the earth) and the gravitational force is keeping the object traveling with it's velocity in circular orbit and preventing it from flying off.

How exactly would the friction force between a car on a flat circular track be an example off centripetal force?

One must understand that a car moves because of the static friction force between the wheels and the ground. If there was no static friction force the car's wheels would spin in place or the ground would be moving underneath it. As a result the static friction force propels the car forward opposite to the direction of tire rotation. Since the car is turning in a circular path it is the static friction force that provides the centripetal acceleration.

REMINDER FBD for Newton's 2nd Law

Reaction Forces are not included in an FBD.

*what's the difference between record payers(phonographs) and CD players?

Record players rotate at constant angular speed while CD players work by picking up the sound at a constant tangential speed. As a result the record rotates constantly while a CD will have to speed up or slow down it's rotation depending on where the laser is above the track. Said another way the angular speed of the CD must vary according to the radial position of the laser.

[ANSWER] How do we know that there is a downwards force R acting at the joint(point O)? what force is missing?

Technically, the force R acting at the joint is the force due to gravity on the humerus [mg(weight) of humerus]. Not pictured is the force due to gravity of the ulna(mg of ulna) but this specific problem/image neglects to include this force though it too is also acting.

[ANSWER] Consider an object of arbitrary shape lying in the x-y plane allowed to rotate about an origin. If the object is divided into a large number of very small particles (each contributing a torque equal to its weight times its lever arm) what would this be equivalent to?

The effect of rotation on the object of the individual particles would be equivalent to a single force of magnitude mg applied at the singular point called the object's center of gravity.

[EXPLANATION] How does curving your body move your center of mass outside your body?

The position of a persons center of mass changes all the time with movement. A standing person has a center of mass inside the body 2/3 up from the ground. If the individual were to lift there hands, the center of mass moves higher because some mass is now redistributed at a higher position. when a person bends over and touches their toes, the center of mass shifts outside the body. Like V shaped objects.

period

Time required for 1 complete revolution

How does rotational kinetic energy change the Work-Energy theorem? What are the new EQUATIONS

Translational & Rotational kinetic energy are now incorporated. Generally speaking, Some kinetic energy is lost as rotational KE.

True or False If a person jumps straight up on a rotating space station he will come straight down.

True... As long as the space station continues to rotate at a constant speed, he will come straight down. This is so because before he jumped he was moving at the same speed relative to the space station and so when he comes down he and the space station will have travelled the same rotational distance so relative to the space station he will come straight down.

Explain conservation of momentum through the example of a spinning figure skater

When the skater pulls her arms and legs close to her body, this reduces their distance from her axis of rotation and hence also reducing her moment of inertia. Because angular momentum is conserved, a reduction in her moment of inertia must increase the angular velocity. Coming out of the spin she needs to reduce her angular velocity so she extends her arms and legs again thereby increasing her moment of inertia and slowing her rotation.

If the applied force is not perpendicular does torque still exists?

Yes torque still exists with the exception of applied forces parallel to the position vector. the trig function sin theta is now used to isolate the component of the torque that is perpendicular to position vector.

tangential speed

[1]the speed at which an object moving in a circular path would move if the force acting on the particle towards the center were to be eliminated. [2] the linear (non-rotational) speed experienced by an object forced to follow a circular path.

tangential acceleration

a measure of how the linear(tangential) speed of an object located a specific distance from the axis of rotation changes with time.

radial acceleration

acceleration pointing towards the radius( or center) radial and centripetal acceleration are used interchangeably

How is an isolated system now defined?

an isolated system is now defined as the state where mechanical energy, linear momentum and angular momentum remain constant

rotational analog of Newton's 2nd law

angular acceleration of an extended rigid object is directly proportional to net torque and inversely proportional to the moment of inertia

when is angular speed positive and negative?

angular speed is positive when theta is increasing (moving counterclockwise) and negative when theta is decreasing (moving clockwise)

angular position

concept is analogous to linear position variable x; angular position is the angle measured in radians at a particular instant in time that the object makes with respect to some fixed reference axis;

Centrifugal force

literally means center fleeing force. From the point of view of the object there seems to be some force that wants the object to fly away outward from the center of the object. however this "feeling" is fictitious and is why centrifugal forces are considered fictitious forces. Your not actually being pushed away from the center though it feels like it. what your actually feeling is a course correction of your path from straight line to circular. Said another way, the inertia from your velocity wants you to keep going straight but the centripetal force makes you follow the circular path. Centrifugal force is a fake and fictional force just like the force you feel when your are in an accelerating car. When a car accelerates you get pushed back into your seat. Nothing is actually pushing you back into your seat. Rather the seat is pushing you forward and you fall backwards.

angular momentum

momentum of rotation; moment of inertia times angular velocity

uniform circular motion

motion of an object moving in a circle at constant speed.

physics behind the high jump and the Fosbury flop

pre-Fosbury, jumpers had to get their entire center of mass over the bar in order to clear it. With the Fosbury flop technique, the jumper gets his entire body over the bar while his center of mass moves under it. To explore this idea further lets break the jumper into three main parts( head, torso/back, & legs. At the initial instance of the jump, the jumpers head is over the bar while his back and legs are not. At the middle instance, his back is over the bar while his head and legs are not. At the last instance, his legs are over the bar but his back and head are not. If you were to average where his center of mass was, it would be under the bar. Since the center of mass can be lower but still allows the jumper to clear the bar, less vertical force is necessary for the jump. In other words an equivalent non-Fosbury vertical force allows a Fosbury jumper to clear a higher bar than the non-Fosbury jumper.

torque

rotational force; any action required to make an object rotate (faster); as force causes acceleration so too does torque cause rotational acceleration. as force causes an object to accelerate so too does torque cause an object's rotation to accelerate

what is the synonym for angular speed?

rotational speed

what's the connection between centripetal acceleration and arc length?

similar triangles the triangle for centripetal acceleration(change in tangential velocity) has the same angular displacement as the triangle for arc length. the triangle between velocity vectors and the triangle between arc length radii have the same angle.

what is the difference between rotational and tangential speed?

tangential (linear) speed increases as that point moves outward away from the center of rotation; the angular speed is the same at every point (no matter how outward(or toward) that point is from the center of rotation).

REMINDER tangential acceleration

tangential acceleration will usually be zero or something we don't particularly worry about unless directly asked. The following are conditions where TA will not be zero. If the velocity speeds(due to changes in angular speed) or if there is a change in the radius during rotation.

unit of torque

the SI unit of torque is the Newton meter (not to be confused with the Joule)

[ANSWERED] QQ8.3 The two rigid objects shown in Figure 8.16 have the same mass, radius and angular speed. If the same braking torque is applied to each, which takes longer to stop (a) A (b) B (c) more information is needed.

the correct answer is (B). while both objects have the same mass, the mass of the ring is concentrated further away from the axis of rotation and thus has a larger torque. So a larger breaking torque will be required to stop that object. See analogous [IMAGE]

Caveat tangential speed

the equation for tangential speed only works when the rotational speed is in rad/s.

arc length

the length of the outer perimeter portion of a circle; the value will be some multiple of the r;

what two conditions must an object satisfy to be in mechanical equilibrium? when are these conditions true?

the net external force and net external torque must equal zero. these conditions are true if the object is moving or rotating at constant speed, or when the object is not moving or rotating at all.

frequency

the number of revolutions(2pi) per second

lever arm

the perpendicular distance from the axis of rotation to a line drawn parallel to a non-perpendicular force.

radian

the primary unit used to measure an angle. it is an alternative to degrees;

Rotational analog of Newton's 1st Law

the rate of rotation of an object doesn't change unless the object is acted on by a net torque.

when solving for net torque what determines whether you add or subtract an individual value of torque within the net torque?

the right-hand rule or counterclockwise torques are added and clockwise torques are subtracted.

when are the average angular speed and instantaneous angular speed equal?

when angular speed is constant