How to find frequency of rotational motion without knowing radius?
#v_1# =3m/s, #v_2# =2m/s, #r_2# =r-10cm
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"Find the limit as x approaches infinity of #xsin(1/x)#?"
I assume that we are referring to a rigid object that rotates at a constant angular frequency #omega# around some axis of rotation.
We know the linear speeds #v_1=3 m/s# and #v_2=2 m/s# at two points #1# and #2# on the rotating solid, and we know that the radii (distance from the axis of rotation) #r_1# and #r_2# at these two points are related as #r_2 = r_1 - 10cm#.
For rigid rotation, it holds for any point on the solid that
#omega = v/r#, where #v# is the linear speed at that point and #r# is the distance from the axis of rotation.
Therefore we know that
#omega = v_1/r_1 = v_2/r_2 = omega#,
which gives that
#v_1/r_1 = v_2/(r_1 - 10 cm)#.
Now we can solve for #r_1# by multiplying both sides by the denominators
#v_1(r_1 - 10 cm)= v_2r_1#,
#r_1 = 10 cm v_1/(v_1-v_2) = 30 cm#.
Using our newfound knowledge of the radius #r_1#, we get that
#omega = v_1/r_1 = (3 m/s)/(30 cm) radians= (3 m/s)/(0.3 m) radians = 10 ((radians)/s)#.
Check that you get the same answer when using #v_2# and #r_2#.
Depends what you're asking.
It looks like there is a typo in your question. If what you are really saying is this: #r_2= r_color(red)(1) -10cm# then:
#omega = 3/r_1 = 2/(r_1 - 10) implies r_1 = 30 " cm" implies omega = 10 "rad/s" #
