What is the period of a book swinging like a pendulum?

In summary, the conversation discusses the calculation of the period of a book acting as a pendulum. The correct moment of inertia is determined using the parallel axis theorem and the distance formula. The correct period is then calculated using the equation T = 2π√[L/g], which yields a value of 0.83 seconds. This is considered the normal and accurate calculation for a simple pendulum.
  • #1
jellyman
4
0
1. In the figure below, a book is suspended at one corner so that it can swing like a pendulum parallel to its plane. The edge lengths along the book face are 28 cm and 19 cm. If the angle through which it swings is only a few degrees, what is the period of the motion?

W0358-N.jpg


2. I=(ML^2)/12
I=(ML^2)/3
T=2∏√I/M*G*dcom


3. I got dcom by using the distance formula and it's .1692 m
Then I tried to using both inertia equations and using length, width. Then I plugged in all the numbers. (multiple attempts)

All wrong answers (.788s, .394s, 1.062s)






Appreciate your help! :)

.
 
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  • #2
You don't have the right moments of inertia.
 
  • #3
Oh.

Then that leaves me with I= M(a2 + b2)/12.

That gave me a period of .476 second which has been marked wrong.
 
  • #4
Yep - that's the wrong moment of inertia as well.
That is for an oblong rotating about it's center.

You book is not rotating about it's center - otherwise it could not act as a pendulum.

Look up: parallel axis theorem.
 
  • #5
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I|| = [itex]\frac{1}{12}[/itex]Mdcom2 + Mdcom2

This doesn't make sense though because it leaves me with an M on the top in the equation for T.EDIT: I asked somewhere else and they used T = 2π√[L/g] and got .83 seconds. This doesn't sound quite right. It's too easy this way!
 
Last edited:
  • #6
$$I_{CM}=\frac{M}{12}(a^2+b^2)$$
##r## is the distance from the corner to the center of the book; by pythagoras: $$r^2=\frac{a^2}{4}+\frac{b^2}{4}$$... therefore, by the parallel axis theorem: $$I=I_{CM}+Mr^2=\cdots$$... you finish up.
 
  • #7
EDIT: I asked somewhere else and they used T = 2π√[L/g] and got .83 seconds. This doesn't sound quite right. It's too easy this way!
That would be pretty normal ... he's modeled the book as a simple pendulum.

For a simple pendulum ##I=ML^2## ... completing the calculations above will tell you how taking the mass distribution into account affects the period.
http://en.wikipedia.org/wiki/Pendulum_(mathematics)#Compound_pendulum
 

What is a book pendulum oscillation?

A book pendulum oscillation is a demonstration of simple harmonic motion using a book as the pendulum. It involves suspending a book from a string or rod and allowing it to swing back and forth.

What factors affect the period of a book pendulum oscillation?

The period of a book pendulum oscillation is affected by the length of the string or rod, the mass of the book, and the gravitational acceleration of the Earth.

How can the period of a book pendulum oscillation be measured?

The period of a book pendulum oscillation can be measured by counting the number of swings in a certain amount of time and calculating the average time per swing. It can also be measured using a stopwatch or a timer.

What is the significance of studying book pendulum oscillations?

Studying book pendulum oscillations can help us understand the principles of simple harmonic motion and the effects of different variables on the period and frequency of oscillation. It also has practical applications in fields such as engineering and physics.

How can the motion of a book pendulum be graphically represented?

The motion of a book pendulum can be graphically represented using a position-time or velocity-time graph. These graphs can show the periodic nature of the oscillation and provide a visual representation of the motion.

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