A simple pendulum of time period 1s and length l is hung from a fixed support at O, such that the bob is at a distance H vertically above A on the ground (Fig. 14.11). The amplitude is θ₀. The string snaps at θ = θ₀ /2. Find the time taken by the bob to hit the ground. Also find distance from A where bob hits the ground. Assume θ₀ to be small so that sine and cosine is equal to 1.

The angular displacement for the pendulum is given as,

----(1)

Where is angular displacement at the given time t and is the maximum angular displacement of the pendulum, and is the angular frequency and t is the time t.

Given:

Time period T of oscillation is = 1s

Length of the pendulum is =

∴

At ,

and,

at the string snaps,

putting the above in equation (1)

now since pendulum moves in the circular motion it must have angular velocity. For that, we differentiate eq. (1)

But

∴

At t=1/6 s

------(2)

But we have a following relation between angular and linear velocity,

, where v is the linear velocity and r is radius and in this case r is the length of the string of the pendulum

∴

Putting the above in equation (2)

We get,

------(3)

∴ the linear velocity at t=1/6 s is given as above.

Now at this stage the string snaps and bob starts a parabolic motion with the above calculated velocity in the tangential direction, thus we need to resolve it into x and y axis component.

Let us look at the vertical component,

Let the bob be at a height from the ground

Velocity along vertical axis,

Now using 2^nd equation of motion, as the bob falls by H₀ distance,

Now solving for t in the above quadratic equation we get,

Now neglecting the term containing , we get,

This is the same time period for which the bob travels along the x-axis after snapping takes place.

And there is no acceleration in the x-axis we can use following equation,

------(4)

Where s is the distance travelled along the x-axis and using equation (3) we can write,

(given )

∴ putting it in equation (4)

And

Since we are given that we can take =1

We get,

∴

Now the horizontal distance from the point A at the time it snaps is,

(since is very)

∴ the distance at which it falls from the point A should be as follows,

The above is the final answer.