T O R Q U E (angular stuff)

Question 1

linear speed vs. rotational speed

Answer 1

linear = distance traveled per time, greater for further out rotating objects

rotational = rotations per unit of time (all points on a disk would have the same speed using rpm)

Question 2

tangential speed

Answer 2

basically linear speed, smth moving along circular path (direction always changing)

Question 3

angular quantities

Answer 3

arc length, radius, and radian angle

Question 4

define a radian, and how to find one given an arc length and radius

Answer 4

the angle at which one arc length is equal to the radius

= arc length / radius

Question 5

how to find linear distance from angle (in radians) and radius

Answer 5

angle times radius

Question 6

convert radians to degrees

Answer 6

2*pi radians = 360 degrees

Question 7

true/false: radians is a unit

Answer 7

false, technically defined as unitless ratio of 2 distances, but it is acceptable to label as “rad” or “radians” to differ from degrees

Question 8

if an object is moving, use the change in angle to determine angular displacement

Answer 8

yes

Question 9

AVERAGE angular velocity

Answer 9

lowercase omega (squishy looking w) = delta theta / delta t

Question 10

INSTANTANEOUS angular velocity

Answer 10

the average for a very short interval

limit as t —> 0 of (delta theta/delta t)
radians/second

Question 11

angular acceleration (lowercase alpha)

Answer 11

a = delta omega/delta t
measured in radians/s^2

Question 12

true/false: w and a are at the same at all points

Answer 12

true, given the object is rigid, as they are properties of the object as a whole

Question 13

prove that linear velocity, v, = r * omega (omega = angular velocity)

Answer 13

m/s = m * (rad/s)
but rad isnt an actual unit! its just a ratio between the arc length and radius!
thus

m/s = m/s

Question 14

connection between linear acceleration a and angular acceleration alpha

Answer 14

a = radius * alpha

same thing with velocity and delta x, just multiple omega by r and delta theta by r respectively

alpha is due to change in speed, acceleration is actually TANGENTIAL acceleration due to speed change

INWARD acceleration exists as well due to velocity changing direction

so a would be a(tangential)

and a(radial) is determined by v^2/r, which equals (r*omega)^2/r, which equals omega^2 * r

Question 15

total acceleration

Answer 15

split into tangential and radial acceleration

radical (a(tan)^2 + a(rad)^2)

using pythagorean theorem

Question 16

true/false: kinematic equations cannot be used on angular problems

Answer 16

false; just plug in the values equal to the angular ones

ex. ample. import java.util.gorrillabookendteacherpeartoothrockeraserradiocomputerrose.;
theta = theta0 + initial omega * t + 1/2alpha*t^2

Question 17

equation for centripetal acceleration

Answer 17

v^2/r

Question 18

what, exactly, is radial acceleration?

Question 19

given a cart, the velocity of the CART itself (ecneter of mass) and linear acceleration of cart itself (center) equals…

Answer 19

V = r*omega of WHEELS

a = r * alpha of WHEELS

Question 20

if a truck is rolling forward at 4.0 m/s,

A. what is the angular velocity of the tires if they have a radius of 70 cm

B. what is the angular acceleration if the truck accelerates 1.5 m/s^2

Answer 20

A.
V = r* omega
4 = .7 * omega
omega = 5.7 rad/sec

B.
a = r * alpha
1.5 = .7 * alpha

alpha = 2.14 rad/sec^2

Question 21

rotation vs. revolution

Answer 21

rotation = spinning around an internal axis

revolution = spinning while “orbiting” an EXTERNAL axis

Question 22

define torque

Answer 22

rotational force

Question 23

symbol for torque

Answer 23

tau (smaller, curlier t)

Question 24

torque equation

Answer 24

tau = radius * force 1

Question 25

true/false: gravity acts at the CENTER of mass

Answer 25

true

Question 26

how would you find the torque of an object with two forces pressing down on it

Answer 26

the net torque (tau1 + tau2) = radius1 * F1 + r2 + F2

Question 27

center of gravity vs. center of mass

Answer 27

generally in everyday objects estimated the same, but technically not the exact same

center of gravity = center of where gravity works

center of mass = center of where mass works

if object is tall enough, gravity pulls harder on the lower parts than the higher parts bcz closer to center of earth, so center of gravity is lower than center of mass

Question 28

Two scales supporot a 3 kg board at each end. the board has a lnegth of 2 meters. a 9 kg object is located 0.70 meters away from the right end of the board. what is the reading on each scale? (how much force is exerted by each scale?)

thinking question:
when calculating the forces on the fulcrum, does chosen fulcrum matter?

Answer 28

no, if everything is in equilibrium, but it could make things a little easer.

for example: mass added to a board with equidistant scales on the left and right sides, with a weight to the right of the center of the board’s mass

use one of the scales as a fulcrum, the force of the scale drops out of the equation and now you only have one unknown

we know that
F1 + F2 = 120
and net torque = 0

choosing the left fulcrum and breaking up the forces as two objects, rather than trying to find the center of mass:
30(1) + 90(1.3) = F2(2)

F2 = 72

F1 + 72 = 120

F1 = 48

Question 29

newtons 2nd law with torque?

Answer 29

linear a = r * alpha

thus F = m * alpha

and since tau = r* F

F = tau/r

so tau = m(net) * r ^2 * alpha

rewritten as:

tau = moment of inertia (I) * alpha

Question 30

moment of inertia

Answer 30

sum of all (mass* radius^2)

ex. (3)(.50)^2 + (4)(.50)^2 = I

Question 31

what do the coefficients in front of the moment of inertia equations tell you

Answer 31

how far the mass generally is from the axis of rotation

Question 32

in which direction is torque positive and negative?

Answer 32

positive in the counterclockwise, negative in the clockwise

generally though probably safest to just say 0.2 in the clockwise or counterclockwise direction

Question 33

the further apart the mass from the axis of rotation… (given the objects youre comparing have the same mass and radius)

Answer 33

the larger the rotational inertia

Question 34

rotational kinetic energy equation

Answer 34

KE(rot) = 1/2 * I * omega^2

Question 35

what is translational energy?

Answer 35

linear energy

Question 36

total kinetic energy of a rolling object

Answer 36

KE(tot) = translational KE + rotational KE

KE(net) =( 1/2 * m * v^2 ) +
( 1/2 * I * omega^2 )

Question 37

sphere falling down ramp slide

Answer 37

MgH = 1/2mv^2 + 1/2I*omega^2
MgH = 1/2mv^2 + 1/2(2/5MR^2)(V/R)^2

R^2 cancels

MgH = 1/2MV^2 + 1/5MV^2

m cancels

gH = 7/10V^2
V = sqrt( gH * 10/7 )

Question 38

does static friction cause energy loss?

Answer 38

no, doesnt do work, so ENERGY isnt lost

Question 39

point mass

Answer 39

all mass approximately concentrated in exact center

Question 40

angular momentum is (conserved/not conserved)

Answer 40

conserved

Question 41

how do figure skaters do the thing

Answer 41

as L = I * omega, decreasing distance = decreasing I, therefore as L is conserved, omega increases

Question 42

law of conservation of angular momentum

Answer 42

Iw = Iw
initial = final

IF the net external torque is 0

Question 43

star with m 8.0 x 10^30 kg collapses into neutron star, original r was 900,000,000 m and spins once per 25 days. if neutron star has radius of 12 km, what is rotational rate

also could you use conservation of energy to solve?

Answer 43

(0.5)(2/5m(900,000)^2) (2pi/25) = (0.5)(2/5m(12)^2)*x
mass cancels out, as does the 1/2 so the m in mr^2 is not necessary

(0.4)*(900,000)^2 * (2pi/25) =
0.4 * 12^2 * x

x = 450,000,000*pi rad/day
which = 225,000,000 rev/day

also no, its an explosion so all the energy is lost

Question 44

can you use conservation of energy on a figure skater?

Answer 44

no, 1/2 * I w^2 = 1/2 I w^2
figure skater is converting glucose and atp into energy, KINETIC energy is not conserved

Question 45

when an ___________ is doing _______, kinetic energy is NOT conserved

Answer 45

internal force

work

Question 46

how do you change the angular momentum of a system

Answer 46

apply an external torque
delta L = torque * delta time

Question 47

formula for angular momentum

Answer 47

L = I*omega

Question 48

given an object rolling (w/o slipping) up a ramp via momentum (no active force), which of the changes would definitely increase distance up the ramp?

a. increase radius of sphere, decrease angular speed, no change to mass
b. decrease linear speed, increase r, decrease m
c. increase r, increase angular speed, increase mass
d. decrease radius, increase speed, decrease mass

Answer 48

in questions like these, look first for the original equations of angular functions/kinematics AS WELL AS your derived stuff (derived equation not included in question).

given that the height is tied directly to linear velocity going up the ramp, and that v = omega * r, a greater radius AND angular speed means an ABSOLUTE GUARANTEED greater height attained. (greater r = greater speed, as we know from far-out points vs near-center points on a spinning disk, and greater rotational speed = greater speed speed of a similar disk, so both means a guaranteed increase in speed)