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Crank
3 years ago
9

The period of a pendulum T is assumed to depend only on the mass m, the length of the pendulum `, the acceleration due to gravit

y g, and the angle of swing θ. By means of dimensional analysis, simplify this problem and express this dependence in non dimensional terms.
Engineering
1 answer:
zzz [600]3 years ago
8 0

Answer:

The expression is shown in the explanation below:

Explanation:

Thinking process:

Let the time period of a simple pendulum be given by the expression:

T = \pi \sqrt{\frac{l}{g} }

Let the fundamental units be mass= M, time = t, length = L

Then the equation will be in the form

T = M^{a}l^{b}g^{c}

T = KM^{a}l^{b}g^{c}

where k is the constant of proportionality.

Now putting the dimensional formula:

T = KM^{a}L^{b}  [LT^{-} ^{2}]^{c}

M^{0}L^{0}T^{1} = KM^{a}L^{b+c}

Equating the powers gives:

a = 0

b + c = 0

2c = 1, c = -1/2

b = 1/2

so;

a = 0 , b = 1/2 , c = -1/2

Therefore:

T = KM^{0}l^{\frac{1}{2} } g^{\frac{1}{2} }

T = 2\pi \sqrt{\frac{l}{g} }

where k = 2\pi

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An Otto cycle engine is analyzed using the air standard method. Given the conditions at state 1, compression ratio (r), and pres
My name is Ann [436]

Answer:

A)  222.58 kJ / kg

B)  0.8897 M^3/ kg

c)  0.7737 m^3/kg

D)  746.542 k

E)  536.017 kj/kg

efficiency = 58% ( approximately )

Explanation:

Given Data :

Gas constant (R) =  0.287 kJ/ kg.K

T1 = 310 k

P1 ( Kpa ) = 100

r = 11.5 ( compression ratio )

rp = 1.95 ( pressure ratio )

A ) specific internal energy at state 1

 = Cv*T1 =  0.718 * 310 = 222.58 kJ / kg

B) Relative specific volume at state 1

= P1*V1 = R*T1 ( ideal gas equation )

V1 = R*T1 / P1 = (0.287* 10^3*310 ) / 100 * 10^3

V1 = 88.97 / 100 = 0.8897 M^3/ kg

C ) relative specific volume at state 2

Applying  r ( compression ratio) = V1 / V2

11.5 = 0.8897 / V2

V2 = 0.8897 / 11.5 = 0.7737 m^3/kg

D) The temperature (k) at state 2

since the process is an Isentropic process we will apply the p-v-t relation

\frac{T1}{T2} = (\frac{V1}{V2}^{n-1}  ) = (\frac{P2}{P1} )^{\frac{n-1}{n} }

hence T2 = 9^{1.4-1} * 310 = 2.4082 * 310 = 746.542 k

e) specific internal energy at state 2

= Cv*T2 = 0.718  * 746.542 = 536.017 kj/kg

efficiency = output /input = 390.3511 / 667.5448 ≈ 58%

attached is a free hand diagram of an Otto cycle is attached below

3 0
3 years ago
A tank with a volume of 8 m3 containing 4 m3 of 20% (by volume) NaOH solution is to be purged by adding pure water at a rate of
lawyer [7]

Answer:

The time necessary to purge 95% of the NaOH is 0.38 h

Explanation:

Given:

vfpure water(i) = 3 m³/h

vNaOH = 4 m³

xNaOH = 0.2

vfpure water(f) = 2 m³/h

pwater = 1000 kg/m³

pNaOH = 1220 kg/m³

The mass flow rate of the water is = 3 * 1000 = 3000 kg/h

The mass of NaOH in the solution is = 0.2 * 4 * 1220 = 976 kg

When the 95% of the NaOH is purged, thus the NaOH in outlet is = 0.95 * 976 = 927.2 kg

The volume of NaOH in outlet after time is = 927.2/1220 = 0.76 m³

The time required to purge the 95% of the NaOH is = 0.76/2 = 0.38 h

4 0
3 years ago
5. Which of the following is false about onStep?
katovenus [111]

The false statement about onStep is: B. The default number of steps per second is 30.

<h3>What is an onStep?</h3>

An onStep can be defined as a computerized telescope goto controller that is designed and developed to <u>animate shapes</u> while using it on a variety of mounting systems such as forks.

<h3>The characteristics of an onStep.</h3>

In Engineering, some of the characteristics that are associated with an onStep include the following:

  • The onStep function can be called without user input.
  • It can be used to animate shapes without user input.
  • It only runs a certain number of times.

In conclusion, the default number of steps per second for onStep isn't 30.

Read more on onStep here: brainly.com/question/25619349

7 0
2 years ago
A consolidation test was performed on a sample of fine-grained soil sample taken from a depth such that the vertical effective s
Scorpion4ik [409]

Answer:

The settlement that is expected is 1.043 meters.

Explanation:

Since the pre-consolidation stress of the layer is equal to the effective stress hence we conclude that the soil is normally consolidated soil

The settlement due to increase in the effective stress of a normally consolidated soil mass is given by the formula

\Delta H=\frac{H_oC_c}{1+e_o}log(\frac{\bar{\sigma_o}+\Delta \bar{\sigma }}{\bar{\sigma_o}})

where

'H' is the initial depth of the layer

C_c is the Compression index

e_o is the inital void ratio

\bar{\sigma_o} is the initial effective stress at the depth

\Delta \bar{\sigma_o} is the change in the effective stress at the given depth

Applying the given values we get

\Delta H=\frac{8\times 0.3}{1+0.87}log(\frac{154+28}{154})=1.04

3 0
3 years ago
List the four processes in the Otto cycle.
OleMash [197]

Answer:

Explanation:

A woman walks due west on the deck of a ship at 3 miyh.The ship is moving north at a speed of 22 miyh.Find the speed and direction of the woman relative to the surface of the water.

7 0
3 years ago
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