Solutions for Capa set 6

 

2. [1pt]
In a hydroelectric power plant, gravitational potential energy is converted to electrical energy by letting the water fall from a great height and then using that energy to turn the electricity generating turbines. How much gravitational potential energy is released when 1291.0 kg of water descends 200 m to the generators?

1.1 Correct, computer gets: 2.53E+06 J

 

 Explanation:

 

Gravitational potential energy is

 

 

Just plug in the numbers

 

 

3.  [2pts]

Cars are tested for safety by performing a series of 'crash tests' where the car (with a crash test dummy behind the wheel) is driven into a cement wall and the entire collision is recorded by a high speed camera to see what happens to the car and the crash dummy during the collision with the wall. Most of the damage (to both the car and to the crash dummy) during a crash is due to the very large forces and associated accelerations occur as the car hits the wall. By performing a large number of such crash tests, automobile engineers can change the design of a car to improve its safety and minimize the damage to occupants during a crash. State whether the following are True or False regarding the physics during a crash.

(Select T-True, F-False, If the first is T and the rest F, enter

TFFFFF).

 

A) By making a car that begins to 'crumple' as the car hits the wall, the car tends not to bounce backwards after the collision and we see that the car has lost most of its kinetic energy by the end of the collision. Therefore, by conservation of energy, much of the kinetic energy must have gone into thermal energy (heat).

B) Some cars are designed to be rigid and bounce backwards very quickly and therefore change their momentum in a very short time when they hit the wall. Other cars are designed to crumple so that the collision takes more time to occur and often avoids bouncing backwards so that they experience a slower change in momentum when they hit the wall. However, the duration of the collision is unimportant, only the change in momentum matters in determining the acceleration.

C) By making the car as rigid as possible, we find that the car bounces backwards after hitting the wall. In bouncing off the wall there is twice the change in velocity of the driver, therefore a larger acceleration, and associated increased chance of injury.

D) A car that bounces off the wall has nearly the same kinetic energy after the collision as it had before the collision. Therefore, by conservation of energy, we expect that the driver could not have been subjected to any significant accelerations.

E) The rigid frame means the car bounces off the wall very well and has nearly the same kinetic energy after the collision as it had before the collision. Therefore, the 'collision energy' (defined on page 86 as the amount of kinetic energy that is

transformed at impact into potential and thermal energy) is mostly thermal energy.

F) By making a car that begins to 'crumple' as the car hits the wall, the collision is found to require more time and the car tends not to bounce backwards after the collision. Therefore, the car's change in momentum must be smaller than if it had bounced back.

 

Correct, computer gets: TFTFFT

Hint: Remember that what causes injury is force on the driver, and this just depends on the driverís acceleration during the crash.

 

Explanation:

A) True, kinetic energy is used to do the work of bending the metal beam, which then is turned into heat

B) False, acceleration is a= (delta p)/(delta t), so a shorter collision time means high acceleration

C) True, a large change in momentum results in a high acceleration

D) False, zero change in energy doesnít imply no accelerations

E) False, bouncing off the wall means that the kinetic energy is the same and because energy is conserved there is no energy to become heat

F) True, because it does not bounce the change in momentum is less.

 

4. [2pt]
In new hybrid gas-electric cars that are just beginning to be sold in the US, as the car gradually slows down, for example when coming to a stop at a red light, the wheels are connected to a generator so that the kinetic energy of the car is converted to electricity which is used to charge up a battery. Then when the car starts up again, the battery drives an electric motor that starts the car, and a small gasoline engine adds a boost to give it more acceleration and to make up for the loss of energy during the stopping to starting conversion. A typical efficiency for the conversion of energy from auto kinetic energy to electrical energy and then back to auto kinetic energy again is 65 percent (e. g. 35 percent of the energy is lost as heat). If the car is going 22 m/s before it starts slowing down, how fast would the car be moving after it had come back up to speed after the stop light? Assume that the gasoline engine was out of gas so it gave no help, and there was no charge in the battery before the car started slowing down.  

 

Correct, computer gets: 1.77E+01 m/s

 

Explanation:

 

This is also an energy problem; however, some of the energy is wasted as heat, so just looking at the initial and final kinetic energy,

 

 

In both cases the energy is purely kinetic so

 

 

Now cancel out the mass from both sides of the equation and solve for vfinal,

 



5. [1pt]

State whether the following are True or False.

(Select T-True, F-False, If the first is T and the rest F, enter

TFFFFF).

 

A) By Bernoulli's equation, water in its streamline can be accelerating only if the water has a decrease in height.

B) You are visiting Marine Land and see a dolphin swimming in a tank. The dolphin being curious comes to look at you

through the glass. You are both at the same level. You are going to feel more pressure than the dolphin because the

water protects him.

C) Water rushing out of a fire hose can provide enough force to knock over a person hit by the flow. Therefore, the

kinetic energy per volume of water is converted to pressure as the water hits the person.

D) Bernoulli's equation is essentially a statement that, for each small volume of water flowing along a streamline,

energy is conserved.

E) Gravity is never an issue when planning water distribution in a building, no matter how tall the building is.

F) Water is in steady flow through a large pipe. Suddenly, the water comes to a place where the pipe is very narrow

and it speeds up to get through the tight spot at higher velocity. Because the water has accelerated, we can state that

the pressure in the narrow region is higher than in the large diameter region.

 

Correct, computer gets: FFTTFF

 

Explanation

 

A) False, because the cross sectional area may increase or decrease causing a change in velocity

B) False, the water is more weight the dolphin must support than the air you must support

C) True, pressure is a force per area and that force does work to knock you over

D) True, yes Bernoulliís equation is base of conservation of energy density

E) False, water tower use gravity

F) False, the high the velocity the lower not high pressure

 

6. [1pt]
Imagine fluid flowing in a horizontal pipe. At one point on a streamline, the pressure is 9890 Pa and the velocity is 2 m/s. At an another point on the same streamline, the pressure is 5270 Pa. What is then the velocity of the fluid?

Correct, computer gets: 3.63868107973205 m/s

Hint: Remember that the density of water is 1000kg/m^3

 

Explanation

 

To solve this problem we use Bernoulliís equation

 

 

Because the pipe is horizontal we can drop the gravity term pgh. Now I will write down the equation for each point along the stream

 

 

Point 1

 

Point 2

 

 

Because this is along a streamline, the Kís are equal

 

 

Now solve for v2

 

 

v2=3.63 m/s

 

7. [1pt]
The National Center for Atmospheric Research has a laboratory located on the Table Mesa in south Boulder. This lab is located high above most of Boulder. To provide reasonable water pressure for the lab, a water tank is located on a ridge 340 m west of the lab and 50 m higher than the lab. The tank has a 3 m2 cross sectional area and there is 49 m of water in it. The tube bringing the water from the tank to the lab is connected to the bottom of the tank. How much water pressure can be provided to the water distribution system in the lab by this tank if the water level of the tank is kept constant by constant monitoring?

Correct, computer gets: 970200 Pa

Hint: Remember that the density of water is 1000kg/m^3

Explanation

 

To solve this problem we use Bernoulliís equation

 

 

because the water is not moving  we can drop the velocity term. I will write down this equation at the point

 

 

Top of the tank

 

                       where h is 50m+49m=99m

 

At NCAR

 

 

 

The Kís are the same, set them equal and solve for Pwater

 

The question is unclear, but the answer is the pressure above atmospheric pressure (gauge pressure)

 

970200Pa

 

 

8. [1pt]
The Hoover dam in Nevada is nearly 250 m high. Water starts at the top of the dam, travels down a large pipe to the bottom of the dam. As it moves down the pipe, its pressure increases enormously. At the bottom of the dam, this pressure is converted to kinetic energy as the water blasts out of the pipe and hits turbine blades at high speed. The turbine blades then rotate to drive an electric generator. What is the speed of the water coming out at the bottom of the pipe according to Bernoulli's equation?

Correct, computer gets: 70 m/s

Hint: Remember that the density of water is 1000kg/m^3

 

Explanation

 

To solve this problem we use Bernoulliís equation

 

 

At the top of the dam

 

 

At the outlet

 

 

The Kís are equal. Solve for v

 

 

 

9. [1pt]

You are opening a bungee jump business where people leap off of a tall bridge. Your typical customer has a mass of 80 kg. What spring constant bungee cord should you use so that the customer's fall stops after the bungee cord has stretched out a distance of 15 m on the first plunge down? The length of the unstretched cord is 210 m, and it is bright red. A bungee cord is just a long spring. Like a spring, the potential energy stored in it is 1/2 the spring constant times the square of the distance it is stretched.

 

In this problem you are dealing with the conversion of energy from gravitational potential energy into to spring energy.  The change in GPE is the  height the customer is lower than when they started, so that is the length of the bungee cord plus the amount it stretches.  So GPE = mg(length+ x) where x is the amount of stretch.  This must be equal to the energy that goes into the stretched cord, which is Ĺ kx^2. 

So mg(length +x) = Ĺ k x^2.  Plug in the numbers given for m, length, and x, and solve for k. 

 

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10. [1pt]

WE WILL THROW THIS PROBLEM OUT!!! (But look at how to solve it anyway!)

A car is heading up a steep road at a high speed, and it runs out of gas. The driver immediately takes it out of gear so they can coast as far up the road as possible. Assume that friction from the road and air resistance are not very important. Which of the following quantities would you need to know to figure out how far they will coast? Mark it T if it is a quantity that you have to know, and F if you could figure out the answer without this quantity. (Select T-True, F-False,   If the first is  T and the rest F, enter

TFFFF). 

 

 A) the slope of the road

 B) the velocity of the car just as it started to coast

 C) the elevation of the car at the start of coasting

 D) the mass of the car alone

 E) the mass of the car and driver together

 

B is the only true answer. Ĺ mv2 = mgh by energy conservation, so v is the only thing you need to predict h.

 

See October 15 class notes

 

11. See Essay questions

 

12. [1pt]

Firefighters are battling a fire in a tall apartment building. The water pressure in the adjacent fire hydrant is about 555000 Pa above atmospheric pressure.

Some firefighters take their hose up the stairwell inside the building. What is the highest level at which they can expect water to flow out of their hose without additional pressure?

 12) 56 [50.4,61.6] m

 

Use Bernoulliís equation to relate change in pressure energy to change in GPE. 550 KPa is converted to pgh, so P = pgh = 550KPa. Plug in and solve for h.

 

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13. [1pt]

Firefighters on the ground begin to spray water upward from their hoses. The water enters the hose traveling slowly at 470000 Pa above atmospheric pressure. How fast will the water be traveling when it leaves the nozzle at atmospheric pressure?

13) 30.6594194335118 [27.5934774901606,33.725361376863] m/s

 

Here P energy is converted into kinetic energy. Change in P energy is 470 KPa = Ĺ Pv2. Plug in for P and solve for v

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14. [1pt]

How high will the water in problem number 13 rise if the firefighters send it straight up?

 

14) 47 [42.3,51.7] m

 

As the water rises the kinetic energy is converted into GPE. So P → kinetic energy → pgh. So you can see that a change in P(470 KPa) = pgh, so h = P/pg = 470 KPa/pg

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15. [1pt]

To boost the water pressure, the firefighters send it sequentially through pumps in two fire engines. Each pump boosts the pressure to get to a total of 1,500,000 Pa above atmospheric pressure. How high will this water rise in hoses carried up the stairs inside the building?

 

15) 153 [137.7,168.3] m

This is just like problem 12. P energy goes into GPE = pgh

So 1500 KPa = pgh

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