The density of glycerin at 25◦c is 1.26201 × 103 kg/m3. its thermal volume expansion coefficient is 4.85 × 10−4 ◦c−1. imagine that i have a 1 cm diameter ball of some material with a density of 1.27000 × 103 kg/m3 and a linear thermal expansion coefficient of 2.25 × 10−4 ◦c−1 sitting in one liter (0.001 m3) of 25◦c glycerin.

(a) how much do i have to change the temperature of the glycerin and the ball to make the ball float (assuming that the glycerin and the ball are always in thermal equilibrium, that both the ball and the glycerin don’t go through a phase change, and that the expansion coefficients don’t change significantly over this temperature range)? give a positive number if you need to increase the temperature, or a negative one if you need to decrease it.

(b) the specific heat of glycerin 2430 j/kg k. how much heat enters the glycerin when i make this temperature change? if heat leaves the glycerin, give your answer as a negative number.