# Question a0c75

Aug 25, 2017

${a}_{\text{c}} = 1008$ ${\text{m s}}^{- 2}$

#### Explanation:

First, let's find the time period of this system.

The string fully rotates $25$ times in $14$ seconds.

Time period is defined as the time taken to complete one full rotation.

Rightarrow "T" = frac(14 " s")(25)

$\therefore \text{T} = 0.56$ $\text{s}$

The system is moving in a circular path, i.e. the acceleration is centripetal.

The equation for centripetal acceleration is ${a}_{\text{c}} = \frac{{v}^{2}}{r}$; where ${a}_{\text{c}}$ is the centripetal acceleration, $v$ is the velocity of the system, and $r$ is its radius (in this case it's the length of the string, i.e. $8$ $\text{m}$).

We still need to find the value of the velocity.

The equation for the velocity of an object travelling in a circular path is $v = \frac{2 \pi r}{\text{T}}$:

Rightarrow v = frac(2 cdot pi cdot 8 " m")(0.56 " s")#

$\therefore v = 89.8$ ${\text{m s}}^{- 1}$

Now, let's find the centripetal acceleration:

$R i g h t a r r o w {a}_{\text{c") = frac((89.8 " m s"^(- 1))^(2))(8 " m}}$

$R i g h t a r r o w {a}_{\text{c") = frac(8064 " m"^(2) " s"^(- 2))(8 " m}}$

$\therefore {a}_{\text{c}} = 1008$ ${\text{m s}}^{- 2}$

Therefore, the acceleration of this system is $1008$ ${\text{m s}}^{- 2}$.