Question

Hydroelectric power plants convert the gravitational potential energy of falling water into electrical power, typically by allowing the water to flow through a pipe called a penstock to rotate a generator located below it. Let the bottom of the penstock be the origin of a Cartesian coordinate system and the point at which the gravitational potential energy is zero.

h = 61 m

a.) Consider a penstock that is vertical and has a height of h = 61 m. How long, t in seconds, does it take water to fall from the top of the penstock to the bottom? Assume the water starts at rest.

b.)How much total mechanical energy in houles does a mass of 1 kg of water have when it reaches the generator?

Answer 1

Answer:

Explanation:

In hydroelectric power , potential energy is converted into electrical energy

height given h = 61 m .

acceleration due to gravity g = 9.8

using the relation

h = ut + 1/2 gt²  , t is time taken to reach the bottom .

61 = 1/2 x 9.8 x t²

t² = 12.49

t = 3.53 s

b )

when a mass of 1 kg of water reaches the bottom , its potential energy will be converted into mechanical energy ( kinetic energy )

potential energy = mgh

m = mass of water = 1 kg

acceleration due to gravity g = 9.8

h = 61 m

potential energy = 1 x 9.8 x 61

= 597.8 J .

mechanical energy = 597.8 J .

Answer 2

The time it takes water to fall from the top of the penstock to the bottom assuming the water starts at rest is 3.53 secs.

The total mechanical energy in Joules that a mass of 1 kg of water have when it reaches the generator is 597.8 Joules

To get the time taken for the water to fall from the top of the penstock to the bottom, we will use the equation of motion as shown:

$S =ut + \frac{1}{2}gt^2$

Given the following parameters

u = 0m/s

g = 9.8m/s²

t is the time taken

S is the distance

Substitute the given parameters into the formula

$61 =0(t) + \frac{1}{2}(9.8)t^2\\61=4.9t^2\\t^2=\frac{61}{4.9}\\t^2= 12.45\\t =\sqrt{12.45}\\t = 3.53 secs$

Hence the time it takes water to fall from the top of the penstock to the bottom assuming the water starts at rest is 3.53 secs.

b) In order to get the total mechanical energy, we must know that the potential energy will be converted into mechanical energy at the bottom. The formula for calculating the potential energy is expressed as:

Potential energy = mgh

Potential energy = 1 × 9.8 × 61

Potential energy = 597.8 Joules

Hence the total mechanical energy in Joules that a mass of 1 kg of water have when it reaches the generator is 597.8 Joules

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