#### Answer

(a) $F_g = 4.90~N$
(b) $T = 2.9~N$
(c) $T = 32.5~N$

#### Work Step by Step

(a) We can find the gravitational force acting on the ball.
$F_g = mg$
$F_g = (0.500~kg)(9.80~m/s^2)$
$F_g = 4.90~N$
(b) We can use the equation for centripetal force to find the tension in the string at the top.
$\sum F = \frac{mv^2}{r}$
$T+mg = \frac{mv^2}{r}$
$T = \frac{mv^2}{r} - mg$
$T = \frac{(0.500~kg)(4.0~m/s)^2}{1.02~m} - (0.500~kg)(9.80~m/s^2)$
$T = 2.9~N$
(c) We can use the equation for centripetal force to find the tension in the string at the bottom.
$\sum F = \frac{mv^2}{r}$
$T-mg = \frac{mv^2}{r}$
$T = \frac{mv^2}{r} + mg$
$T = \frac{(0.500~kg)(7.5~m/s)^2}{1.02~m} + (0.500~kg)(9.80~m/s^2)$
$T = 32.5~N$