How much $g$ can the wire be accelerated?

Question

A wire can withstand the tension $T_{max} = M_{max}\cdot g$. If we hang on an object with the mass of $m = \frac{M_{max}}{10}$ at the end of the wire, how much $g$ can the wire be accelerated? ($g$ is gravitational force)

I've tried equate them

$T_{max} = M_{max}\cdot g = \frac{M_{max}}{10}$

and found the answer as $9g$. However, I'm not sure.

Regards!

Answer 1

HINT

We have that (neglecting the wire mass)

• initial tension: $T_0=mg=\frac{M_{max}}{10}g$
• increment in tension due to an up acceleration $a$: $\Delta T=ma=\frac{M_{max}}{10}a$

then the total tension in the wire is

$$T=T_0+\Delta T=\frac{M_{max}}{10}(g+a)$$