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 "T" = 0.56# #"s"#

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

The equation for centripetal acceleration is #a_("c") = frac(v^(2))(r)#; where #a_("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# #"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)("T")#:

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

#therefore v = 89.8# #"m s"^(- 1)#

Now, let's find the centripetal acceleration:

#Rightarrow a_("c") = frac((89.8 " m s"^(- 1))^(2))(8 " m")#

#Rightarrow a_("c") = frac(8064 " m"^(2) " s"^(- 2))(8 " m")#

#therefore a_("c") = 1008# #"m s"^(- 2)#

Therefore, the acceleration of this system is #1008# #"m s"^(- 2)#.