Deriving C=mv2/r and a=v2/r: Mass, Gravity & Acceleration

[fcb]

I don't know if this is the right equation but in a question where it sakes the acceleration is 60%of the cars weight. Why does the mass cancel each other out so in the end you don't need to know the mass?

Sorry I am typin this on my phone so I am sorry for the spelling errors


Thanks

Edit: some won understand what I mean. When they say the acceration is a certain percentage of the weight. Why do you multiply it but gravity. Essestially using the formula v2/r

Can someone derive it for me. How does v2/r come about? How is there mass in the first place and why do you multiply it with the gravity constant of earth?

 

[Maybe_Memorie]

I'm not entirely sure what you're asking..
From what I can gather, you want to know how to derive a=v2/r and why weight is equal to mg, correct?

 

[clombard1973]

I don't know if this is the right equation but in a question where it sakes the acceleration is 60%of the cars weight. Why does the mass cancel each other out so in the end you don't need to know the mass?

Sorry I am typin this on my phone so I am sorry for the spelling errors


Thanks

Edit: some won understand what I mean. When they say the acceration is a certain percentage of the weight. Why do you multiply it but gravity. Essestially using the formula v2/r

Can someone derive it for me. How does v2/r come about? How is there mass in the first place and why do you multiply it with the gravity constant of earth?

Let's start with where they get V²/r from.
If you want to know the distance around some part of a circle given an angle theta you can find it by using S = theta*r, take the derivative of each side to get the tangential velocity going around the circle:

dS/dt = (dtheta/dt)*r = V​

and dtheta/dt is the angular velocity, w. So,

V = w*r​

Now you need units of acceleration on each side, this will be the centripetal acceleration that we find having units of m/s², so square each side and divide by r yielding:

V²/r = w²r​

Now V²/r has units of m²/m*s² = m/s², which are the correct units. So the centripetal acceleration is given by two formulas, those being:

V²/r = w²r​

OK, now you have been given the centripetal acceleration as 60% of the cars weight so the centripetal force acting on the car which acts to pull it inward like a satellite orbiting the Earth must be: 0.6mg, now set that equal to the centripetal acceleration times the mass of the car and you should be all set:

0.6mg = m*V²/r​

This is where the mass cancels on each side. How much of a centripetal force there is acting on a rotating body is independent of the mass of that body and is dependent only on the tangential speed it is moving at and the distance from the origin of the circle it travels along. So,

(5.88*r)^0.5 = V​

Tangential Velocity of Car
where 5.88 = 0.6*g = 0.6*9.8 = 5.88

Or,

0.6mg = m*w²r​

5.88 = w²r
(5.88/r)^0.5 = w​

The angular velocity of the car

So if you need the tangential velocity of the car use:

(5.88*r)^0.5 = V​

Or if you need the angular velocity (how many times per second does the car make a complete circle) then use:

(5.88/r)^0.5 = w​

Hope this helps.
Craig

 

[fcb]

Let's start with where they get V²/r from.
If you want to know the distance around some part of a circle given an angle theta you can find it by using S = theta*r, take the derivative of each side to get the tangential velocity going around the circle:

dS/dt = (dtheta/dt)*r = V​

and dtheta/dt is the angular velocity, w. So,

V = w*r​

Now you need units of acceleration on each side, this will be the centripetal acceleration that we find having units of m/s², so square each side and divide by r yielding:

V²/r = w²r​

Now V²/r has units of m²/m*s² = m/s², which are the correct units. So the centripetal acceleration is given by two formulas, those being:

V²/r = w²r​

OK, now you have been given the centripetal acceleration as 60% of the cars weight so the centripetal force acting on the car which acts to pull it inward like a satellite orbiting the Earth must be: 0.6mg, now set that equal to the centripetal acceleration times the mass of the car and you should be all set:

0.6mg = m*V²/r​

This is where the mass cancels on each side. How much of a centripetal force there is acting on a rotating body is independent of the mass of that body and is dependent only on the tangential speed it is moving at and the distance from the origin of the circle it travels along. So,

(5.88*r)^0.5 = V​

Tangential Velocity of Car
where 5.88 = 0.6*g = 0.6*9.8 = 5.88

Or,

0.6mg = m*w²r​

5.88 = w²r
(5.88/r)^0.5 = w​

The angular velocity of the car

So if you need the tangential velocity of the car use:

(5.88*r)^0.5 = V​

Or if you need the angular velocity (how many times per second does the car make a complete circle) then use:

(5.88/r)^0.5 = w​

Hope this helps.
Craig

That was so helpful. Honestly,Thanks so much

 

[fcb]

Quick question. Why is it to the power of 0.5??