What is the mathematical proportionality for centripetal acceleration?

In summary, a group of individuals have been conducting an experiment on Cent acceleration and have determined that T is proportional to M^-.35, m^.53, and r^.5 using mathematical analysis and plotted data. The constant for this relationship can be determined from the intercepts of logarithmic graphs.
  • #1
bayan
203
0
Hi everyone!

We have been doing a n experiment about Cent acceleration.

We had a formula which was UgM= 4Pie^2rm/T^2

Then we have T being ptoportional to M, R and m

Can someone help me to figure out the proportionality?

It is to be determined using maths although I have got some result (not using maths, only from our observation of graphs and making the LOG graph to find gradiant..) I have got it to be
T proportional to M^-.35
T proportional to m^.53
T proportional to r^.5


Is there anyone that can help me to fing what the mathematical proportionality would be?

Also any work out would be apriciated to explain.
 
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  • #2
Your plotted data has given you empirical exponents for the dependence of T on each of three variables. I assume your data was collected by allowing only one of the three to vary at any time. When you say "T proportional to M^-.35" You are saying T is some constant times M^-.35, but that constant may depend on the other two variables that you are holding fixed. Similar reasoning appplies to the other two variables. Putting them all together gives you

T = Constant*M^-.35*m^.53*r^.5

The value of the constant can be deduced from the intercepts of logarithmic graphs you plotted.

Your data does not quite agree with your starting formula, but it is not terribly far off.
 
  • #3


The mathematical proportionality for centripetal acceleration is given by the formula a = v^2/r, where a is the centripetal acceleration, v is the tangential velocity, and r is the radius of the circular motion. This means that as the tangential velocity increases or the radius decreases, the centripetal acceleration will also increase.

In your formula, UgM = 4π^2rm/T^2, T is proportional to M, r, and m. This can be rewritten as T^2 ∝ Mrm. Using the proportionality relationship for centripetal acceleration, we can see that T^2 ∝ v^2/r. Since T^2 is proportional to v^2 and r, we can say that T is proportional to v and r separately. This means that T ∝ v and T ∝ r.

To find the exact mathematical proportionality, we can use dimensional analysis. Since T is proportional to v and r, we can write T = kv^x r^y, where k is a constant and x and y are the exponents that we need to determine.

To find the value of x, we can look at the units. T has units of time (s), v has units of velocity (m/s), and r has units of length (m). So, we can write the equation as s = (m/s)^x (m)^y. Simplifying this, we get x + y = 1.

To find the value of y, we can use the fact that T ∝ v^2 and r ∝ v^2. This means that T = kv^2 and r = kv^2. Substituting these into our original equation, we get T^2 ∝ (kv^2)^2 (kv^2)^2. Simplifying this, we get T^2 ∝ k^2v^4. Since T^2 is proportional to v^4, we can say that y = 4.

Substituting these values into our original equation, we get T = kv^1 r^4. Simplifying this, we get T = kv^1 r^2. This means that the mathematical proportionality for centripetal acceleration is T ∝ v^1 r^2, or written in a more familiar form, T = krv^2. This shows that T is directly proportional to the radius squared and
 

What is centrepetal acceleration?

Centrepetal acceleration is the acceleration of an object moving along a curved path towards the centre of the circular motion.

What causes centrepetal acceleration?

Centrepetal acceleration is caused by the centripetal force, which is directed towards the centre of the circular motion.

How is centrepetal acceleration calculated?

Centrepetal acceleration can be calculated by dividing the square of the velocity by the radius of the circular motion.

What is the difference between centrepetal acceleration and tangential acceleration?

Centrepetal acceleration is directed towards the centre of the circular motion, while tangential acceleration is directed along the tangent to the circular path.

What are some real-world examples of centrepetal acceleration?

Some examples of centrepetal acceleration include the motion of planets orbiting the sun, a car turning a corner, and a rollercoaster moving along a curved track.

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