To determine the tension force in the string AB that lifts a block of mass [tex]\(\frac{5}{8} \, \text{kg}\)[/tex] with an upward acceleration of [tex]\(2 \, \text{m/s}^2\)[/tex], we can follow these steps:
1. Calculate the weight of the block:
The weight is the force due to gravity acting on the mass of the block. It can be calculated using the formula:
[tex]\[
\text{Weight} (W) = m \cdot g
\][/tex]
where [tex]\(m\)[/tex] is the mass and [tex]\(g\)[/tex] is the acceleration due to gravity.
Given:
[tex]\[
m = \frac{5}{8} \, \text{kg}
\][/tex]
[tex]\[
g = 9.8 \, \text{m/s}^2
\][/tex]
Therefore:
[tex]\[
W = \left(\frac{5}{8}\right) \cdot 9.8 = 6.125 \, \text{N}
\][/tex]
2. Calculate the net force acting on the block:
The net force is the sum of the forces exerted on the block. When the block is lifted with an acceleration [tex]\(a\)[/tex] upward, three forces act on it: the tension in the string [tex]\(T\)[/tex], the weight [tex]\(W\)[/tex] of the block, and the net upward force due to the acceleration.
According to Newton's second law:
[tex]\[
F_{\text{net}} = m \cdot a
\][/tex]
where [tex]\(a\)[/tex] is the acceleration.
Given:
[tex]\[
a = 2 \, \text{m/s}^2
\][/tex]
Therefore:
[tex]\[
F_{\text{net}} = \left(\frac{5}{8}\right) \cdot 2 = 1.25 \, \text{N}
\][/tex]
3. Determine the tension [tex]\(T\)[/tex]:
The tension force [tex]\(T\)[/tex] in the string must counteract both the weight of the block and the additional net force due to the block's upward acceleration.
Using the relation:
[tex]\[
T = W + F_{\text{net}}
\][/tex]
Therefore, substitute the values of [tex]\(W\)[/tex] and [tex]\(F_{\text{net}}\)[/tex]:
[tex]\[
T = 6.125 + 1.25 = 7.375 \, \text{N}
\][/tex]
So, the tension force in the string AB is [tex]\(\boxed{7.375 \, \text{N}}\)[/tex].
