Question

The diameter of the bottom of the mug drink in the figure below is 6 cm. The diameter of the opening
9 cm . When you roll the cup on a flat surface it rolls in a circle. Which end ,
bottom or the opening of the cup rolls faster ? How much faster?

Answer 1

What's your guess?

Answer 2

the bottom, But I want help how to calculate how much faster

Answer 3

Or no, I need help how to calculate it

Answer 4

Sorry, I'm not sure on how you can calculate it exactly. Look into angular velocity and angular frequency, though.

Answer 5

I though the bottom as well and as for calculating it I think you divide the two numbers. as to which order I'm not sure if you divide 9 by 6 you get 1.5 if you divide it by 6 by 9 you'll just get the simplification of 2/3 or 0.6 repeating

Answer 6

however I'm not too sure

Answer 7

But maybe its okey to answear 2/3 faster?

Answer 8

I guess and I hope

Answer 9

if you can, can you help me with my question?

Answer 10

Thank you anyways @iPwnBunnies :)

Answer 11

try to trace out the two circles for the bottom and top of the mug
Imagine the bottom and top have wet paint on them (black paint). and they go around the circle

Answer 12

in the time it takes to make one revolution, the bottom traces out a smaller circle than the top circle.

Answer 13

so clearly the top end goes around the circle faster, right?

Answer 14

yes!

Answer 15

And is there a way to calculate how much faster?

Answer 16

yes

Answer 17

v^2=a*r?

Answer 18

If u knew the radius, how would you calculate it?

Answer 19

@perl is the opening 1.5x faster than the bottom?

Answer 20

or 67% faster?

Answer 21

if we imagine we are looking at the bottom of the cup

Answer 22

we can use the formula
linear velocity = radius * angular speed (in radians / unit time)

Answer 23

we know that angular speed must be the same, since the bottom of the cup and the top of the cup are making revolutions synchronised

Answer 24

\[v_1*r_1=\omega_1 \\ v_2*r_2=\omega_2 \\ \text{but } \\ \omega_1 = \omega_2 \\ \text{so} \\v_1*r_1=v_2*r_2 \\ \text{hence by algebra} \\ \frac{v_1}{v_2}=\frac {r_1}{r_2}\]

Answer 25

so
v1 / v2 = 9/6 = 1.5

v1 = 1.5 * v2

so the top of the cup is moving 1.5 times faster than the bottom of the cup

Answer 26

So I had it right! I just couldnt described it like you! Perl, You are amazing! I wish you all the best! Thanks alot