Circling steel balls

$T=\dfrac{mv²}{R} \rightarrow T'=T+64=\dfrac{m(3v)²}{R+dR}$ *

Dividing the 2 equations:

$\dfrac{T+64}{T}=9 \therefore \frac{64}{T}=8 \therefore T=8,,$

We know, from the centripetal force, that, if $F$ is the force of tension, then $F \propto v^2$ , since the tension is the source of the centripetal acceleration. Therefore, if $v$ is multiplied by $3$ , $F$ is multiplies by $9$ .

Let the original tension force, in Newtons, be $T$. Then, we have: $9T = T + 64 \implies T = 8,$ so our answer is $\boxed {8} \, \text{N}.$

using centripetal force formular, F=mv2/r, i use 2 in place of squaed let the initial force be F=mv2/r _equation 1, then the new force, F+64=(m(3v)2)/r _equation 2, subtracting equation 1 from 2, we get 64=9mv2/r - mv2/r=8mv2/r, therefore, mv2/r=64/8=8N, remember mv2/r=F which is the original tension force.

Let T be the original tension in the string.

mv^2 / r=T

m(3v)^2 / r=T + 64

(9mv^2) / r = T + 64 , We also know that mv^2/r=T and we can sub that into this equation which gives,

9T = T + 64

8T=64

T=8

Therefore the original tension was 8N.

Tension in the string will be (m $v^{2}$ )/r. If T is the original tension, then T = (m $v^{2}$ )/r and the new tension T' = (m $(3v)^{2}$ )/r = 9T. It is given that 9T-T=64. Thus 8T = 64 => T = 8.

The tension causes the centripetal acceleration of the ball. The magnitude of centripetal acceleration experienced by the ball is $a_{c} = \frac{v^{2}}{r}$ . Thus, the initial tension experienced by the ball is $T_{i} = \frac{m \times v^{2}}{r}$ .

When the velocity is tripled, the centripetal acceleration is now nine times the original centripetal acceleration, i.e. the tension experienced by the ball is now $T_{f} = \frac{9\times m \times v^{2}}{r}$ .

The increase in tension is thus $T_{f} - T_{i}= \frac{8\times m \times v^{2}}{r} = 64 N$ .

The initial tension is thus $T_{i} = \frac{8\times m \times v^{2}}{r} \times \frac{1}{8} = \frac{64}{8} = 8 N$ .

In the original condition the tension force cancels with the centripetal force so T = m v^2 / R then the velocity is increased by a factor of 3 so the velocity now becomes 3v. as the velocity increases, we are told that the Tension force also increases by 64 N so T + 64 = m (3v)^2 / R T + 64 = 9 m v^2 / R from the first equation we get T + 64 = 9T 64 = 8T T = 8 N