Instantaneous Acceleration for Uniform Circular Motion

[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

v_o = d/t
v_o = (2*r*pi) / t
v_o = (2*30*pi) / 60
v_o = 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 + (v_o)²
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?

 

[kuruman]

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

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 $\vec a=\omega^2 R~(-\hat y)=-3.3\times 10^{-3}(\mathrm{m/s^2})~\hat y.$