Question

a bowling ball is thrown at an initial speed of Vo. It has mass M, radius R and coefficient of friction uk. Find the velocity at which the ball starts pure rolling.

Answer 1

i did this using two methods but they gave me 2 different answers. So, can someone please asses it?

Answer 2

In a past exam, they asked the question without giving coefficient of friction. if the coefficient is given, i can use method 1 but without it, ill have to use method 2 which gives me a wrong answer?!

Answer 3

@surry99

Answer 4

@Pompeii00

Answer 5

please, note that befor pure rolling, energy of your sphere is decreased, due to friction. Initial energy is kinetic energy, since sphere behaves as like a punctiform mass with a speed V_0. So I think initial energy is:
\[E=\frac{ 1 }{ 2 }MV _{0}^{2}\]
where M is the mass of your sphere

Answer 6

do you agree?

Answer 7

@physo

Answer 8

what do you think about that?

Answer 9

Please, note that I can not give you the solution directly, since the Code of Conduct!

Answer 10

that equation describes the linear kinetic energy

Answer 11

so i think it might complicate it

Answer 12

part of that energy will lost due to friction. Now if S is the space path by our sphere before pure rolling, the residual energy will be:
\[\frac{ 1 }{ 2 }MV _{0}^{2}-\mu _{k}gS\]
do you agree?

Answer 13

would it not be mgus?

Answer 14

oops sorry you are right!
\[-\mu _{k}MgS\]

Answer 15

ok...now?

Answer 16

=0.5Iw^2?

Answer 17

I think we have to apply differential calculus

Answer 18

namely second cardinal equation of dinamic of rigid body

Answer 19

we have that the torque of external force is due to friction force, and its component is:
\[-\mu _{k}MgR\]
are you agree?

Answer 20

so the second cardinal equation of rigid body will be:
\[I \frac{ d \omega }{ dt }=-\mu _{k}MgR\]
where omega is angula frequency of our sphere
@physo

Answer 21

yes

Answer 22

but note that u is not given

Answer 23

and I is the moment of inertia of our sphere

Answer 24

sorry u is the friction coefficient?

Answer 25

sorry u is the friction coefficient?

Answer 26

yes

Answer 27

i don't know how to write the symbol

Answer 28

from your text I read that
\[\mu _{k} \]
is known

Answer 29

Sorry, but if we want to solve your problem it is necessary to consider u_k like a known quantity

Answer 30

anyway acceleration of the center of mass of our sphere is:
\[a _{G}=-\mu _{k} g\]
so its speed is:
\[v _{G}=V _{0}-\mu _{k}g t\]

Answer 31

integrating the above differential equation you will get, with our initial condition over omega:
\[\omega(t)=-\frac{ 5}{ 2} \frac{ \mu g t}{R} \]

Answer 32

true. I'll give you the exact question

Answer 33

now I ask you, a simple question, namely what is the equation of the speed of contact point of our spere with the motion plane?

Answer 34

part c

Answer 35

if you see the attachment with my worked solution, you'll see that i wrote exactly what you did. You don't have to waste your time explaining what i already did.

Answer 36

that's is another question I think!

Answer 37

I'm sorry I have to go to sleep, because in my country, namely Italy we are beyond midnight

Answer 38

oh im really sorry

Answer 39

but ill give you a medal for your efforts

Answer 40

here is link to the problem solved in two different manners
http://www.feynmanlectures.info/exercises/bowling_ball_rolling.html

Answer 41

Take a look and I would be happy to discuss with you later

Answer 42

wow. that just made everything clear. thx a lot.

Answer 43

you are welcome

Answer 44

ok! good morning sirs, I'm here.
I think if no frction exists between sphere and plane of motion,then sphere never starts to roll without sliding. I think friction is the necessary condition so that your sphere begins to roll without sliding