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Q.80

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

The analysis of a simple pendulum assumed that the mass was a particle, with no size. A realistic pendulum is a small, uniform sphere of mass and radius at the end of a massless string, withbeing the distance from the pivot to the center of the sphere.

a. Find an expression for the period of this pendulum.

b. Suppose, typical values for a real pendulum. What is the ratio, whereis your expression from part a and is the expression derived in this chapter?

a. the expression for the period of this pendulum

b. The ratio

See the step by step solution

Step by Step Solution

Step 1: Given information (part a)

mass of the sphere

Radius of the sphere

length of the pendulum

acceleration due to gravity

Step 2: Explanation (part a)

If the mass of a simple pendulum is held and displaced slightly to one side through a small angle , then allowed to oscillate about which is shown in figure.

The kinetic energy at would be minimum, but maximum at as a result of the the energy loss is moving from towards .

The maximum kinetic energy gained

At the potential energy is maximum as a result of the height the bob is raised when displaced, the maximum potential energy

is constant.

The force that tend to move it towards the centre is given by

By comparing this with then

To determine the period of oscillation:

Since the angle through which the bob is displaced is small the arc becomes approximately a straight line

Then,

Therefore equation becomes

that is, acceleration,

but

and

hence

Therefore, the expression for the time period of oscillation is given by,

Step 3: Given information (part b)

mass of the sphere

radius of the sphere

length of the pendulum

Step 4: Explanation (part b)

The time period of oscillation of a pendulum is given by,

Plugging the given values in the above equation, we get

The time period of simple pendulum, which was already derived in this chapter is given by,

Plugging the values in the above equation, we get

Therefore the ratio of is given by

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