Question

A stone is tied to a string and whirled at constant speed in a horizontal circle. The speed is then doubled without changing the length of the string. Afterward the magnitude of the acceleration of the stone is:

Answer 1

Answer:

Afterward the magnitude of the acceleration of the stone is four times as great

Explanation:

A stone tied to a string and whirled at a constant speed in a horizontal circle will have a centripetal acceleration given by

$a = \frac{v^{2} }{r}$

Where $a$ is the centripetal acceleration

$v$ is the speed

and $r$ is the radius of the circle

Here, the radius of the circle is the length of the string.

Now, from the question

The speed is then doubled without changing the length of the string,

Let the new speed be $v_{2}$, that is

$v_{2} = 2v$

and without changing the length of the string means radius $r$ is constant.

To determine the magnitude of the acceleration of the stone afterwards,

Let the new acceleration be $a_{2}$.

Then we can write that

$a_{2} = \frac{v_{2}^{2} }{r}$

From

$a = \frac{v^{2} }{r}$

$v = \sqrt{ar}$

Recall that

$v_{2} = 2v$

∴ $v_{2} = 2\sqrt{ar}$

Now, we will put the value of $v_{2}$ into

$a_{2} = \frac{v_{2}^{2} }{r}$

Then,

$a_{2} = \frac{(2\sqrt{ar}) ^{2} }{r}$

$a_{2} = \frac{4ar }{r}$

$a_{2} = 4a$

The new magnitude of the acceleration of the stone is four times the initial acceleration.

Hence,

Afterward the magnitude of the acceleration of the stone is four times as great

Answer 2

On doubling the speed, with unchanged length of string, the  magnitude of acceleration of stone will become four times greater than that of initial value.

As per the given problem, as the stone tied to a string and whirled at a constant speed in a horizontal circle will have a centripetal acceleration. Then the expression for the centripetal acceleration is given by,

$a =\dfrac{v^{2}}{r}$ ...............................................................(1)

here,

v is the constant speed of stone.

r is the radius of horizontal circle. And this radius is nothing but equivalent to the length of string. That is,

r = L

If the speed of stone is doubled, then the new speed is,

v' = 2v

And new acceleration of stone from equation (1) is given as,

$a' =\dfrac{v'^{2}}{L}\\\\\\a' =\dfrac{(2v)^{2}}{L}\\\\\\a' =\dfrac{4v^{2}}{L}\\\\a'=4a$

Thus, we can conclude that on doubling the speed, with unchanged length of string, the  magnitude of acceleration of stone will become four times greater than that of initial value.

Learn more about the centripetal acceleration here:

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