Chapter 11: Rotational Dynamics and Static Equilibrium

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What is the relationship between torque and radius?

-torque increases with the distance r from tne axis of rotation to the force

What is the general definition of torque?

Torque = r (F sin theta)

What type of torque does radial force produce?

-zero torque

What are the units of torque?

N * m

What does a torque applied to an object give rise to?

-an angular acceleration

The greater the torque applied to an object, the...?

-greater its angular acceleration

If torque acts alone, what is the sign of its value if it causes a counterclockwise angular acceleration?

Torque > 0

If torque acts alone, what is the sign of its value if it causes a clockwise angular acceleration?

Torque < 0

Torque is the...?

-rotational analog to force

If we apply a force to a mass, in what direction will it accelerate in?

-in the same direction as the force

If we apply a torque to an object, in what direction will the rotational acceleration act in?

-in the same direction as the torque

What is newton's second law in the rotational domain?

τ = Iα

What is I?

-a measure of angular inertia
-(how stubborn an object is to changes in its rotational motion, α)

If something has a large angular inertia, how resistant is it to changes in its α?

-very resistent

What three factors does rotational inertia depend upon?

-leverage
-the force itself
-orientation

If Fnet = 0, what is the acceleration of an object?

-when Fnet equals zero, acceleration is zero

What are the two options in terms of the movement of the object?

-the COM stays at rest
-the COM has constant speed in a straight line

If τnet = 0, what is the rotational acceleration of an object?

-when τnet = 0, the rotational acceleration of the object is zero

What are the two options in terms of the movement of the object?

-the object stays not spinning
-the object spins at a constant rate in a constant direction (either cw or ccw)

When does the tendency for a force to cause a rotation increase?

-when distance from the pivot point increases

What is the relative distance of the force from the pivot point known as?

-leverage

What is the broadest formula for the size of torque?

τ = rF

What is the only type of force that this equation includes?

-tangential force

What are the units of τ?

N * m

What is r?

-distance from the pivot to the force application point

If r increases, what is the effect on τ?

-τ increases

Correspondingly, what is the effect on α?

-α increases

What torque do radial forces produce?

-zero torque

What is the general definition of torque?

τ = r (Fsinθ)

What does this formula take into account?

-the component of a force that is tangential and that which is radial

Which force is the only one that needs to be accounted for?

-the tangential force

Why?

-it is the only part of the force that is useful for rotation

When τ > 0, in what direction does an object accelerate in?

CCW

When τ < 0, in what direction does an object accelerate in?

CW

What are N * m equal to?

-joules

In that case, why aren't joules used to refer to the size of torque?

-the size of torque is scalar, not a vector
-->therefore, the term joule cannot be used

If force is 90 degrees to the radius, how much of the force is tangential?

-100%

Therefore, what is Fsinθ equal to?

-1

Therefore, we achieve __________ torque.

-maximum

If force is 0 or 180 degrees to the radius, how much of the force is tangential?

-0%

Therefore, what is Fsinθ equal to?

-0

How much torque does this produce?

-zero torque

If you are working with objects that are balanced with a single support force, where must that support force be located?

-at the COM

What are the two limitations of the balance method?

1. the balance method is not helpful if you have more than one support force
2. if objects tip, they are not supported

You are given two rotating systems. In the first system, two spherical masses are positioned far from the axis of rotation. IN the second system, the two spherical masses are positioned near the axis of rotation. If the hanging blocks are released simultaneously from rest, is it observed that (a) the block at left lands first, (b) the block at right lands first, or (c) both blocks land at the same time?

-block at right lands first

Why?

-the moment of inertia of the system at right is less than that of the system at left because the movable masses are closer to the axis of rotation Since the angular acceleration is inversely proportional to the moment of inertia, the system at the right has the greater angular acceleration

What two conditions must be met for an extended object to be in static equilibrium?

-the net force acting on the object must be zero
-the net torque acting on the object must be zero

What is the relationship between these to conditions?

-there is no relationship between the two
-->the two conditions are completely independent of each other
-->satisfying one does not guarantee that the other is satisfied

When completing static equilibrium problems in this class, how many equations will we need to use? What type?

-2: one force equation and one torque equation

How many unknowns will we need to solve for?

-2

What are the 4 steps to solving a static equilibrium problem?

1. choose a "rotational system"
2. identify external forces on the system
3. pick a pivot
4. solve for up to two unknowns using two equations

Why must we only identify external forces?

-because only external forces will influence the system

If the extended object which your system is centered upon is massless, how many mg values will be in your equation?

-only 1 (if there is only one object resting on the extended object)

If the object which your system is centered upon is not massless, how many mg values will be in the equation?

-2 values (if there is only one object resting on the extended object)

What should be the location of the pivot that you choose?

-the pivot should be at a location where there are the most number of forces

Why?

-this makes calculations easier because the torque at this location will always be equal to zero

Why?

-because the r value (leverage) will be equal to zero

When a system exists where net torque is equal to zero, what are the respective sizes of two active torques?

-the two active torques are equal in magnitude but opposite in direction

What does rotational inertia help us to link?

-τ net with α

What is rotational inertia a measure of?

-rotational inertia is a measure of a mass's stubbornness to change in α

What are the three factors that rotational inertia depends upon?

1. mass
2. mass distribution
3. axis of rotation

What is the relationship between mass and I?

-they are directly proportional to each other

What is the effect on I if m increases?

-I increases

Will the object therefore be more or less resistant to changes in its acceleration?

-more resistant

What is the relationship between I and mass distribution?

-as mass becomes more distributed, rotational inertia increases

Will the object therefore be more or less resistant to changes in its acceleration?

-more resistant

What is the relationship between I and axis of rotation?

-the larger your axis of rotation is from the COM, the greater your rotational inertia is

Will the object therefore be more or less resistant to changes in its acceleration?

-more resistant

Can you have a negative inertia value?

-no

Why not?

-just as you cannot have a negative mass, you cannot have a negative inertia

Why is the inertia of a thin ring greater than the inertia of a solid disk of the same mass?

-because the mass of the thin ring is distributed further from the radius

Why is the inertia of a bar that rotates around its COM less than the inertia of a bar of the same mass that rotates around a point further from its COM of the same mass?

-because the axis of rotation of the second bar is further from its COM

Two forces produce the same torque. Does it follow that they have the same magnitude? Explain.

-no
-torque depends on both the magnitude of the force and on the distance from the axis of rotation (or leverage) at which it is applied
-a small force can produce the same torque as a large force if it is applied farther from the axis of rotation

A tightrope walker uses a long pole to aid in balancing. Why?

-the long pole helps to distribute the mass of the tightrope walker, leading to a greater rotational inertia
-with a greater rotational inertia leads to a small angular acceleration, giving the walker more time to "correct" his/her balance

Give an example of a system in which the net torque is zero but the net force is nonzero.

-a force applied radially to a wheel produces zero torque, even though the net force is nonzero

Is the normal force exerted by the ground the same for all four tires on your car?

-no
-in mos cars, the engine is located in the front. Thus the car's center of mass is not in the middle of the car, but is closer to the front end
-this means that the force exerted on the front tires is greater than the force exerted on the rear tires

Give two everyday examples of objects that are in static equilibrium.

-i am in static equilibrium as a sit in my chair
-so is the building where I have my physics class

Stars form when a large rotating cloud of gas collapses. What happens to the angular speed of the gas cloud as it collapses?

-the angular speed of the dust cloud increases, just like a skater pulling in his/her arms
-->due to a decreased distribution of mass

Is it possible to change the angular momentum of an object without changing its linear momentum?

-yes, most definitely
-->imagine turning on a ceiling fan: this increases the fan's angular momentum without changing its linear momentum

A uniform disk stands upright on its edge, and rests on a sheet of paper placed on a tabletop. If the paper is pulled horizontally to the right, does the disk rotate clockwise or counterclockwise about its center?

-the disk rotates counterclockwise, as one would expect from a force exerted to the right on the bottom of the disk

Does the center of the disk move to the right, to the left, or does it stay in the same location?

-the center of the disk moves to the right, since that is the direction of the net force exerted on the disk

A disk and a hoop (bicycle wheel) of equal radius and mass each have a string wrapped around their circumferences. Hanging from the strings, halfway between the disk and the hoop, is a block of mass m. The disk and the hoop are free to rotate around their centers. When the block is allowed to fall, does it stay on the center line, move toward the right, or move toward the left? Explain.

-the block moves to the left
-the reason is that the hoop, with the larger moment of inertia, has the smaller angular acceleration
-since less string unwinds from the hoop, the block moves in that direction

A puck on a horizontal, frictionless surface is attached to a string that wraps around a pole of finite radius. As the puck moves along the spiral path, does its linear speed increase, decrease, or stay the same? Explain.

-stays the same
-Since the string is always at right angles to the motion of the puck, the tension does no work on it
-->therefore, the linear speed of the puck (v) remains constant

Does its angular speed increase, decrease, or stay the same?

-increases
-as the string becomes shorter, its angular speed (w = v/r) increases

A beetle sits near the rim of a turntable that is rotating without friction about a vertical axis. The beetle now begins to walk toward the center of the turntable. As a result, does the angular speed of the turntable increase, decrease, or stay the same?

-as the beetle walks towards the axis of rotation, the COM moves closer towards the axis of rotation
-as a result, inertia decreases
-since inertia decreases, angular acceleration increases, and angular speed increases!

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