Period=1.4 seconds
gravitational acceleration=9.8 m/s^2
length of string=.4865m
angle is 3 degrees
acceleration from that angle is=.5129 m/s^2
change in bobs height from release to bottom of arc=.000667m
bobs speed at bottom of arc=.1143 m/s
so i done all the questions and calculations for one above but i not sure how to do this one. Any help is appreciated.
Describing a pendulum as a harmonic oscillator requires the use of the small angle approximation. What is the magnitude of the difference between the actual acceleration (without using the small angle approximation) and the approximate acceleration (using the small angle approximation)? The answer is a very small number. As for all other numerical answers, provide your answer with 4 significant digits.
Assume that the angle is 3 degrees = 0.05235988 radians.
The actual acceleration of the pendulum in the direction of motion, when A = 3 degrres, is g sin A =
9.81 * 0.05233596 = 0.51341573 m/s^2
The small angle approximation says that the acceleration is g * A(in radians) =
0.51376504 m/s^2
Take the difference and keep 4 siginificant figures. It would also be useful to express it as a percentage.
i tried .0003493 and -.0003493 but both are wrong where am i messing up exactly?
Ellie,
If you had redone the calculations, you would have found that the value for the small approximation is 0.51365039886193 instead of 0.51376504 which makes the difference 0.0002347, and after rounding to 4 significant figures as required by the question, will become 0.0002 m/s².
See also my previous responses, corrected for parts d and e at:
http://www.jiskha.com/display.cgi?id=1257393909
To find the magnitude of the difference between the actual acceleration and the approximate acceleration, we need to calculate both accelerations and then subtract the approximate acceleration from the actual acceleration.
The actual acceleration can be calculated using the centripetal acceleration formula:
acceleration_actual = (4π² * length_of_string * sin(angle)) / (period²)
Given that the length_of_string is 0.4865m and the angle is 3 degrees (which is approximately 0.0524 radians), we can calculate the actual acceleration.
acceleration_actual = (4π² * 0.4865 * sin(0.0524)) / (1.4²)
Next, we can calculate the approximate acceleration using the small angle approximation, which states that for small angles, sin(angle) is approximately equal to the angle in radians.
acceleration_approximate = (4π² * length_of_string * angle) / (period²)
Using the same values as before, we can calculate the approximate acceleration.
acceleration_approximate = (4π² * 0.4865 * 0.0524) / (1.4²)
Finally, to find the magnitude of the difference between the actual acceleration and the approximate acceleration, we subtract the approximate acceleration from the actual acceleration and take the absolute value.
difference = abs(acceleration_actual - acceleration_approximate)
Now, you can substitute the given values into the formulas and calculate the difference. Make sure to round your answer to 4 significant digits as requested.