Physics Ch. 7

if an object rotates about its end (axis on end of object) what happens to the inertia?

the moment of inertia is greater. it's further from the axis of rotation

the CG will be closer to what side?

the more massive object/particle

suppose a 67 kg mass is at the origin and a 28-g mass is at x = 49 cm. What's the center of gravity?

(0*.67 + 49*28)/(67-28) Xcg = +1.4E-01 m

a pivot point on a see saw would represent what if the two sides were balanced?

center of gravity/balance point

ex of constant angular accel: the crankshaft in a race car goes from rest to 3000 rpm in 2.0s. a) what's the crankshaft's angular accel (in rad/s^2) b) how many revolutions does it make during that time interval? first, find ___, second find ___, third find ___

1- alpha, 2- delta theta (radians), 3-revolutions a) a = delta w/delta t = (3000*2pi/60-0)/(2.0) =157 rad/s^2 b) delta theta = (wi)(delta t) + 1/2a(deltat)^2 = 0 + 1/2(157)(2.0)^2 = 314 rad =314/(2pi) = 50 rev = +5.0E+01 rev (1 rev = 360 deg = 2pi radians) (1 rad = 1 rev/2pi)

ex: a wheel's rotating at 15 rpm. What's its angular speed in rad/s? Explain how to do the problem

15 rev/min = 15 * 2pi rad/60s = 1.6E+00 rad/s

because its sing fives directional information, w (omega) can be considered a(n) ___

1D vector

1 revolution = 360 deg = ___ radians 180 deg = ___ rad 90 deg = ___ rad and so on

2pi pi pi/2

for a force being applied at some point, what must be the angle between the force and its radial line for the resulting torque to be maximum?

90 degrees

if w (omega) is positive, rotation is ___ and if it's negative, rotation is ___.

CCW CW

the sign of omega tells whether the object is rotating ___ or ___

CCW CW

suppose we had instead a solid cylinder of same diameter and moment of inertia. What would the mass be?

I = 1/2MR^2 0.19 = 1/2M(0.65/2)^2 M = +3.6E+00 kg

a bicycle rim has a diameter of .65 m and moment of inertia, measured about its center, of .19 kg*m^2. What's the mass of the rim?

I = MR^2 0.19 = M(.65/2)^2 M = +1.8E+00 kg

formula for moment of inertia for a collection of particles:

I = m1r1^2 + m2r2^2 + m3r3^2...= sum of miri^2

for a point mass m, a distance r from an axis, its moment of inertia about that axis is ___

I = mr^2

recall, for a rotating object, we also defined its period of rotation T = 2pi(r)/v. Using v = (omega)r, it gives us yet another relationship: ___ used

T = 2pi/w radians

ex: what's the period of the motion for a wheel rotating at 15 rpm? Explain how to solve

T = 2pi/w = 2pi/(15*2pi/60) = 1/(15/60) = +4.0E+00s 2pi/((15(2pi)/60) = 1/(15/60) (2pi cancels)

a rotating wheel has a moment of inertia of 7.7*10^-4 kg*m^2. What net torque must be applied to the wheel for its angular acceleration to be 48 rad/s? explain how to solve

Tnet = Ia = (7.7*10^-4)(48) = +3.7E-02 N*m

Nt's 2nd law applied to rotational motion in formula form: to what is this analogous in linear motion?

Tnet = Ia Fnet = ma

1D formula for the location of the center of gravity:

Xcg = (x1m1 + x2m2)/(m1 + m2)

Newton's second law for rotation states:

an object that experiences a net torque Tnet about an axis of rotation undergoes an angular acceleration a = Tnet/I where I is the moment of inertia of the object about the rotation axis

linear v circular motion: position, m (x) = ___(___) velocity, m/s (vx) = ___(___) acceleration, m/s^2 (ax) = ___(___)

angle, radian (theta) angular velocity, rad/s (omega) angular acceleration, rad/s^2 (alpha)

a net torque is the cause of ___

angular acceleration

In the absence of a net torque, the object has zero ___ so it either doesn't ___ or it rotates with constant ___

angular acceleration rotate angular velocity

the absolute value of angular velocity is called ___, but they often use the same symbol, depending on context

angular speed

for a rotating, solid (rigid) object, such as a rotating disc, all points have the same ___ and ___, but their speeds would depend on how far the point is from the ___

angular speed, period axis of rotation

the angular displacement through which the particle moves each second: denoted:

angular velocity w (omega)

the net weight force can be considered as acting at a single point know as the ___

center of gravity

if in UCM (constant speed), the angular velocity would be ___ and its angular acceleration would be ___

constant zero

T or F: the formula for the moment of inertia of a circular object wouldn't depend on whether the object was hollow/solid

false

for a solid disk rotating about its axis at a constant angular speed, points ___ the center move faster

farther

Points at a greater radii from the axis of rotation travel ___. Why?

faster to keep up

What causes an object to have angular acceleration? The answer has something to do with ___, but is more specifically due to the ___ that results from the applied forces.

force(s) net torque

Mathematically, the center of gravity represents the average location of the ___ acting on the body or the ___ of the object

gravitational force balance point

the coefficient of the moment of inertia formula is a measure of what?

how difficult it is to rotate something. The greater the coefficient, the harder it is to rotate

the center of a wheel is what?

its center of gravity

moment arm is also called ___

lever arm

the ability of a force to cause an angular acceleration depends of the ___ and ___ of the force and where it's acting relative to the ___

magnitude, direction pivot point

___ alone is the inertia associated with linear motion

mass

The formula for rotational inertia depends on the ___ and how the mass is ___ relative to the axis

mass distributed

for a wheel rotating counterclockwise and slowing down, what's it's angular acceleration?

negative

Tnet = Ia Tnet = ___ I = a =

net torque about axis of rotation moment of inertia of object about axis of rotation angular acceleration

-torque has SI units of ___ -torque is measured relative to some ___/___ -torque is a(n) ___ in the sense that it's positive or negative based on the direction of rotation that it tries to cause. -torque is ___ if it tries to cause rotation in a counterclockwise direction and ___ if it tries to cause rotation in a clockwise direction.

newton-meters (N*m) axis/point vector positive, negative

for UCM the formula for angular velocity is: SI units for angular velocity:

omega = delta theta/delta t rad/s

Points at a greater radii from the axis of rotation travel faster to keep up. This is the ___ to orbital motion. why?

opposite satellites aren't rigid and don't have to stay in a line

when considering applying force at the end of a wrench, the ___ component doesn't contribute to torque, but the ___ component does

parallel perpendicular

-maximum positive torque is produced by a force ___ to radial line -pushing straight toward pivot point exerts ___ torque -positive torque tries to rotate object ___ about pivot -pulling straight out from pivot exerts ___ torque -a negative torque tries to rotate object ___ about pivot -maximum negative torque is produced by a force ___ to radial line

perpendicular zero CCW zero CW perpendicular

(r_I_) extends from ___ to ___ and is ___ to line of action

pivot, line of action perpendicular

ex: what's the magnitude of the max torque a 16-N force can apply to a .35-m-long bar w/ pivot point at one of is ends and the force applied to its other end?

r = .35, (F_I_) = 16 .35*16 = +5.6E+00 N*m

the line from pivot point to the point where force is applied

radial line

use ___ for all formulas of this chapter in which angles have to be measured

radians

recall that one way to indicate how fast an object's rotating is by specifying its ___. Another way is to specify its ___.

revolutions per time (rpm) angular velocity

angular acceleration is negative when a rigid body's: (2)

rotating counterclockwise and slowing down rotating clockwise and speeding up

angular acceleration is positive when a rigid body's: (2)

rotating counterclockwise and speeding up rotating clockwise and slowing down

what is rolling without slipping a combination of?

rotation and translation

more descriptive name for I: what are its units?

rotational inertia kg*m^2

arc length is denoted by ___

s

relationship between arc length, radius, and angle in radians is shown by ___ formula -notice this formula gives the correct formula for the ___ of a circle by setting theta equal to 2pi: c = 2pi(r)

s = r(theta) circumference

would a solid cylinder or a cylindrical hoop have a smaller moment of inertia? why?

solid cylinder- the points are closer to the center/axis of rotation

fig 7.33: a tangential force exerts a torque on the particle and causes the ___ and causes the particle's ___ to change. The tension causes the ____ and causes the particle's ___ to change

tangential acceleration, speed centripetal acceleration, direction

since w = mg, the location of the center of gravity is also where what is?

the center of mass

the perpendicular component is perpendicular to what?

the radial line

The ability to cause an angular acceleration is called the ___ produced by/ due to the ___

torque force

compare 3 ways an object can move: ___- mov't from point A to point B ___-stationary axis of rotation

translation rotation

true or false: any force that points either directly towards/away from pivot point can't produce a torque about the pivot point

true

deduce the linear relationship between the linear speed at which the center of a rolling object is moving and the angular speed of the object by considering the uniform motion case: (formula) what formula is this super similar to?

v = wR v = linear speed of center of rolling object R = radius w = angluar speed of rolling object rotational motion, just different r. No differentiation on formula sheet

the ("regular") speed of a point a distance r from the axis of rotation is related to its angular speed through the equation: ___ must be used!!

v = wr radians

angular acceleration is a(n) ___ in the sense that: it's ___ if delta w > 0 and its ___ if delta w < 0

vector positive negative

What is the net torque of the center of gravity about that point?

zero

alternate formula for magnitude of torque: the magnitude of the torque due to a force F with a moment arm r_I_

|t| = (r_I_)F

the magnitude of the torque due to a force with perpendicular component F_l_ acting at a distance r from the pivot is given by the formula:

|t| = r(F_I_)