# Instantaneous Acceleration for Uniform Circular Motion

• Error
In summary: The post above this one in the current thread referenced above provides a clear explanation of the method used to determine the instantaneous acceleration. The answera = (velocity at 11.999s - velocity at 12.001s) / 12.001 - 11.999given in the old thread is a reasonable approximation for the instantaneous acceleration. However, the method used in the current thread referenced above produces the exact answer.
Error

## Homework Statement

A normally functioning clock has a radius of 30cm. What is the average acceleration in the first 15 seconds if the second hand starts at 12? What is the instantaneous acceleration when the second hand is at 12?

## Homework Equations

v = d/t

a = Δv / Δt
limit as Δt approaches 0

## The Attempt at a Solution

vo = d/t
vo = (2*r*pi) / t
vo = (2*30*pi) / 60
vo = 3.14 cm/s

v = d/t
v = (2*r*pi) / t
v = (2*30*pi) / 60
v = 3.14 cm/s

For average velocity between 0 - 15 seconds:

a = Δv / Δt

Δv² = v² + (vo
Δv = 4.44 cm/s [S 45° W]

a = 4.44 / 15
a = 0.30 cm/s² [S 45° W]

I'm not too sure how to find the instantaneous acceleration. I have no idea what a limit is or how to use it.

a = Δv / Δt
a = (velocity at 11.999s - velocity at 12.001s) / 12.001 - 11.999

I'm not sure if this is anything close to being right?

I am posting here being fully aware that this thread is more than 8 years old and that the OP has not been seen for almost 7 years. I believe that unanswered threads in the "Related Threads ..." section of the Introductory Physics Homework threads are of no help to anyone. For that reason I am posting a complete solution for the benefit of all. This old thread is related to the current thread https://www.physicsforums.com/threads/instantaneous-acceleration-from-coordinates.809726/.

Lest it be thought that I am violating forum rules by doing so, I point out that the span of eight years most likely exceeds the statute of limitations from the date of the original post. I base this conclusion on my assessment that such violation of PF rules can be no more egregious than the following felonies that have a statute of limitations of only five years in the state of Texas where I reside. (https://brettpodolsky.com/criminal-...e-of-limitations-expired-for-your-texas-crime)

Under Tex. Code Crim. Proc. Ann. Art. 12.01(4), specific felony offenses have a five-year statute of limitations from the date on which the crime was allegedly committed, including:
• Robbery
• Theft
• Kidnapping
• Burglary
• Injury to a disabled or elderly individual (unless punishable as a first-degree felony under Texas Penal Code Section 22.04)
• Endangering/abandoning a child
Now onto the solution.

Using conventional Cartesian coordinates, we write the initial and final velocities as ##\vec v_i=\omega R~\hat x~;~\vec v_f=-\omega R~\hat y.## The average acceleration is$$\vec a_{avg.}=\frac{\vec v_f-\vec v_i}{\Delta t}=-\frac{\omega R} {\Delta t}(\hat x+ \hat y).$$ With ##\omega=\dfrac{2\pi}{60} \mathrm{s^{-1}}## and ##R=0.3~ \mathrm{m}##, the expression evaluates to ##\vec a_{avg.}=-2.1\times 10^{-3}(\mathrm{m/s^2})(\hat x+\hat y).## OP's answer
Error said:
a = 0.30 cm/s² [S 45° W]
is correct.

OP did not realize that the instantaneous acceleration at the 12:00 position is centripetal and given by ##\vec a=\omega^2 R~(-\hat y)=-3.3\times 10^{-3}(\mathrm{m/s^2})~\hat y.##

## 1. What is instantaneous acceleration for uniform circular motion?

Instantaneous acceleration for uniform circular motion is the rate of change of velocity at any given point along the circular path. It is a vector quantity and is always directed towards the center of the circle.

## 2. How is instantaneous acceleration calculated for uniform circular motion?

Instantaneous acceleration for uniform circular motion can be calculated using the formula a = v2/r, where a is the acceleration, v is the velocity, and r is the radius of the circle.

## 3. Is instantaneous acceleration constant for uniform circular motion?

No, instantaneous acceleration for uniform circular motion is not constant. This is because the velocity and direction of the object are constantly changing as it moves along the circular path, resulting in a changing acceleration.

## 4. What is the relationship between instantaneous acceleration and centripetal acceleration?

Instantaneous acceleration and centripetal acceleration are closely related. Centripetal acceleration is a special case of instantaneous acceleration, and it is always equal in magnitude to instantaneous acceleration but in the opposite direction.

## 5. Can instantaneous acceleration be negative for uniform circular motion?

Yes, instantaneous acceleration can be negative for uniform circular motion. This occurs when the velocity and direction of the object are changing in opposite directions, resulting in a negative acceleration value.

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