T O R Q U E (angular stuff) Flashcards
linear speed vs. rotational speed
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)
tangential speed
basically linear speed, smth moving along circular path (direction always changing)
angular quantities
arc length, radius, and radian angle
define a radian, and how to find one given an arc length and radius
the angle at which one arc length is equal to the radius
= arc length / radius
how to find linear distance from angle (in radians) and radius
angle times radius
convert radians to degrees
2*pi radians = 360 degrees
true/false: radians is a unit
false, technically defined as unitless ratio of 2 distances, but it is acceptable to label as “rad” or “radians” to differ from degrees
if an object is moving, use the change in angle to determine angular displacement
yes
AVERAGE angular velocity
lowercase omega (squishy looking w) = delta theta / delta t
INSTANTANEOUS angular velocity
the average for a very short interval
limit as t —> 0 of (delta theta/delta t)
radians/second
angular acceleration (lowercase alpha)
a = delta omega/delta t
measured in radians/s^2
true/false: w and a are at the same at all points
true, given the object is rigid, as they are properties of the object as a whole
prove that linear velocity, v, = r * omega (omega = angular velocity)
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
connection between linear acceleration a and angular acceleration alpha
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
total acceleration
split into tangential and radial acceleration
radical (a(tan)^2 + a(rad)^2)
using pythagorean theorem
true/false: kinematic equations cannot be used on angular problems
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
equation for centripetal acceleration
v^2/r
what, exactly, is radial acceleration?
given a cart, the velocity of the CART itself (ecneter of mass) and linear acceleration of cart itself (center) equals…
V = r*omega of WHEELS
a = r * alpha of WHEELS
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
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
rotation vs. revolution
rotation = spinning around an internal axis
revolution = spinning while “orbiting” an EXTERNAL axis
define torque
rotational force
symbol for torque
tau (smaller, curlier t)
torque equation
tau = radius * force 1
true/false: gravity acts at the CENTER of mass
true
how would you find the torque of an object with two forces pressing down on it
the net torque (tau1 + tau2) = radius1 * F1 + r2 + F2
center of gravity vs. center of mass
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
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?
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
newtons 2nd law with torque?
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
moment of inertia
sum of all (mass* radius^2)
ex. (3)(.50)^2 + (4)(.50)^2 = I
what do the coefficients in front of the moment of inertia equations tell you
how far the mass generally is from the axis of rotation
in which direction is torque positive and negative?
positive in the counterclockwise, negative in the clockwise
generally though probably safest to just say 0.2 in the clockwise or counterclockwise direction
the further apart the mass from the axis of rotation… (given the objects youre comparing have the same mass and radius)
the larger the rotational inertia
rotational kinetic energy equation
KE(rot) = 1/2 * I * omega^2
what is translational energy?
linear energy
total kinetic energy of a rolling object
KE(tot) = translational KE + rotational KE
KE(net) =( 1/2 * m * v^2 ) +
( 1/2 * I * omega^2 )
sphere falling down ramp slide
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 )
does static friction cause energy loss?
no, doesnt do work, so ENERGY isnt lost
point mass
all mass approximately concentrated in exact center
angular momentum is (conserved/not conserved)
conserved
how do figure skaters do the thing
as L = I * omega, decreasing distance = decreasing I, therefore as L is conserved, omega increases
law of conservation of angular momentum
Iw = Iw
initial = final
IF the net external torque is 0
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?
(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
can you use conservation of energy on a figure skater?
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
when an ___________ is doing _______, kinetic energy is NOT conserved
internal force
work
how do you change the angular momentum of a system
apply an external torque
delta L = torque * delta time
formula for angular momentum
L = I*omega
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
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)
