The mass of an elevator is 500 kg. Calculate the tension in the cable when the elevator is (i) stationary, (ii) ascending with acceleration 2 m/s², (iii) descending with acceleration 2 m/s². Take g = 9.8 m/s².

Here are the calculations for the tension in the elevator cable under the given conditions.

Given: Mass of elevator, $m = 500 \text{ Kg}$ Acceleration due to gravity, $g = 9.8 m/s^2$ Acceleration of elevator, $a = 2 m/s^2$

The forces acting on the elevator are the tension $T$ (upwards) and its weight $mg$ (downwards). According to Newton's second law, the net force $\sum F = ma$. Taking the upward direction as positive: $\sum F = T - mg$ So, $T - mg = ma$. Rearranging for tension: $T = mg + ma$

i) Stationary When the elevator is stationary, its acceleration $a = 0 m/s^2$. Step 1: Substitute the values into the tension formula. $T = mg + m(0)$ $T = mg$ Step 2: Calculate the tension. $T = (500 Kg)(9.8 m/s^2)$ $T = 4900 N$ The tension in the cable when the elevator is stationary is $\boxed{4900 N}$.

ii) Ascending with an acceleration of $2 m/s^2$ When the elevator is ascending, the acceleration $a = +2 m/s^2$ (positive because it's upwards). Step 1: Substitute the values into the tension formula. $T = mg + ma$ $T = m(g+a)$ Step 2: Calculate the tension. $T = 500 Kg (9.8 m/s^2 + 2 m/s^2)$ $T = 500 Kg (11.8 m/s^2)$ $T = 5900 N$ The tension in the cable when the elevator is ascending is $\boxed{5900 N}$.

iii) Descending with the same acceleration ($2 m/s^2$) When the elevator is descending, the acceleration $a = -2 m/s^2$ (negative because it's downwards). Step 1: Substitute the values into the tension formula. $T = mg + ma$ $T = m(g+a)$ Step 2: Calculate the tension. $T = 500 Kg (9.8 m/s^2 + (-2 m/s^2))$ $T = 500 Kg (9.8 m/s^2 - 2 m/s^2)$ $T = 500 Kg (7.8 m/s^2)$ $T = 3900 N$ The tension in the cable when the elevator is descending is $\boxed{3900 N}$.