The knot at the junction is in equilibrium under the influence of four forces acting on it. The F force acts from above on the left at an angle of α with the horizontal. The 5.7 N force acts from above on the right at an angle of 50◦ with the horizontal. The 6.2 N force acts from below on the right at an angle of 44◦ with the horizontal. The 6.7 N force acts from below on the left at an angle of 43◦ with the horizontal.1. What is the magnitude of the force F?
2. What is the angle a of the force F in the figure above?
Answers
(a) The magnitude of the force F acting on the knot is 5.54 N.
(b) The angle α of the force F is 54.4⁰.
The given parameters:
- F force at α
- 5.7 N force at 50⁰
- 6.2 N force at 44⁰
- 6.7 N force at 43⁰
The net vertical force on the knot is calculated as follows;
The net horizontal force on the knot is calculated as follows;
From the trig identity;
The angle α of the force F is calculated as follows;
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The knot is in equilbrium, so there is no net force acting on it. Starting with the unknown force and going clockwise, denote each force by F₁, F₂, F₃, and F₄, respectively. We have
F₁ + F₂ + F₃ + F₄ = 0
Decomposing each force into horizontal and vertical components, we have
F cos(180º - α) + (5.7 N) cos(50º) + (6.2 N) cos(-44º) + (6.7 N) cos(-137º) = 0
F sin(180º - α) + (5.7 N) sin(50º) + (6.2 N) sin(-44º) + (6.7 N) sin(-137º) = 0
Recall that cos(180º - x) = - cos(x) and sin(180º - x) = sin(x), so these equations reduce to
F cos(α) ≈ - 3.22 N
F sin(α) ≈ 4.51 N
(1) Recall that for all x, sin²(x) + cos²(x) = 1. Use this identity to solve for F :
(F cos(α))² + (F sin(α))² = F ² ≈ 30.73 N² → F ≈ 5.5 N
(2) Use the definition of tangent to solve for α :
tan(α) = sin(α) / cos(α) ≈ 1.399 → α ≈ 126º
or about 54º from the horizontal from above on the left of the knot.
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A frictionless pendulum is made with a bob of mass 12.6 kg. The bob is held at height = 0.650 meter above the bottom of its trajectory, and then pushedforward with an initial speed of 4.22 m/s. What amount of mechanical energy does the bob have when it reaches the bottom?
Answers
Answer:
h = 16.9 m
Explanation:
When a ball is thrown upward, its velocity gradually decreases, until it stops for a moment, when it reaches the maximum height, while its height increases. Thus, the law conservation of energy states in this case, that:
Kinetic Energy Lost by Ball = Potential Energy Gained by Ball
(0.5)m(Vf² - Vi²) = mgh
h = (0.5)(Vf² - Vi²)/g
where,
Vf = Final Speed of Ball = 0 m/s (Since, ball stops for a moment at highest point)
Vi = Initial Speed of Ball = 18.2 m/s
g = acceleration due to gravity = - 9.8 m/s² ( negative for upward motion)
h = maximum height the ball can reach = ?
Therefore, using values in the equation, we get:
h = (0.5)[(0 m/s)² - (18.2 m/s)²]/(-9.8 m/s²)
h = 16.9 m
(b) the exergy destroyed during this process.
Answers
A) The exergy of the refrigerant at the initial and final states are :
- Initial state = - 135.5285 kJ
- Final state = -51.96 kJ
B) The exergy destroyed during this process is : - 1048.4397 kJ
Given data :
Mass ( M ) = 5 kg
P1 = 0.7 Mpa = P2
T1 = 60°C = 333 k
To = 24°C = 297 k
P2 = 100 kPa
A) Determine the exergy at initial and final states
At initial state :
U = 274.01 kJ/Kg , V = 0.034875 m³/kg , S = 1.0256 KJ/kg.k
exergy ( Ф ) at initial state = M ( U + P₂V - T₀S )
= 5 ( 274.01 + 100* 10³ * 0.034875 - 297 * 1.0256)
≈ - 135.5285 kJ
At final state :
U = 84.44 kJ / kg , V = 0.0008261 m³/kg, S = 0.31958 kJ/kg.k
exergy ( ( Ф ) at final state = M ( U + P₂V - T₀S )
= -51.96 kJ
B) Determine the exergy destroyed
exergy destroyed = To * M ( S2 - S1 )
= 297 * 5 ( 0.31958 - 1.0256 )
= - 1048.4397 KJ
Hence we can conclude that A) The exergy of the refrigerant at the initial and final states are : Initial state = - 135.5285 kJ, Final state = -51.96 kJ and The exergy destroyed during this process is : - 1048.4397 kJ
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Final answer:
Exergy of refrigerant-134a at initial and final states is obtained from property tables and by multiplying the mass of the refrigerant with its specific exergy at each state. The difference in exergy between the two states represents the exergy destroyed.
Explanation:
To solve the given question, we need the property values of
refrigerant-134a
at the initial and the final states.
At an initial state of 0.7 MPa and 60°C, the specific exergy for refrigerant-134a can be obtained from property tables which are standard in thermodynamics textbooks. Same for the final state at 0.7 MPa and 24°C, the specific exergy can be obtained from the same property tables.
The exergy of the refrigerant at the initial and the final states can be calculated by multiplying the mass of the refrigerant with its specific exergy at each state.
Exergy destruction during this process can be calculated using the relation between exergy change and exergy destruction. The exergy change of a system between initial and final states is equal to the difference of the exergy of the system at final and initial states.
Based on the second law of thermodynamics, the difference in exergy should be equal to the exergy destroyed during the process.
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B. Ultraviolet light waves
C. Infrared Waves
D. Microwaves
Answers
The heat emitted from anything is carried in the form of infrared waves. (C)
Answers
Answer:
0.5 lambda(wavelength)
Explanation:
We know that
The first harmonic for both side open ended pipe is
L= 1/2lambda
So L = 0.5*wavelength
Answers
Answer:
29.5 m/s
Explanation:
Volumetric flowrate = (average velocity of flow) × (cross sectional area)
Volumetric flowrate = 0.111 liters/s = 0.000111 m³/s
Cross sectional Area of flow = πr²
Diameter = 0.00579 m,
Radius, r = d/2 = 0.002895 m
A = π(0.002895)² = 0.0000037629 m²
Velocity of flow = (volumetric flow rate)/(cross sectional Area of flow)
v = 0.000111/0.0000037629
v = 29.5 m/s
Given Information:
diameter of the nozzle = d = 5.79 mm = 0.00579 m
flow rate = 0.111 liters/sec
Required Information:
Velocity = v = ?
Answer:
Velocity = 4.21 m/s
Explanation:
As we know flow rate is given by
Flow rate = Velocity*Area of nozzle
Where
Area of nozzle = πr²
where
r = d/2
r = 0.00579/2
r = 0.002895 m
Area of nozzle = πr²
Area of nozzle = π(0.002895)²
Area of nozzle = 2.6329x10⁻⁵ m²
Velocity = Flow rate/area of nozzle
Divide the litters/s by 1000 to convert into m³/s
0.111/1000 = 1.11x10⁻⁴ m³/s
Velocity = 1.11x10⁻⁴/2.6329x10⁻⁵
Velocity = 4.21 m/s
Therefore, the water exit the nozzle at a speed of 4.21 m/s
Answers
Answer:
Force that the output plunger applies to the car; F2 = 3888N
Explanation:
For a hydraulic device, the relationship between the force and the area using Pascal's principle is;
F1/A1 = F2/A2
Where;
F1 is force applied to the input piston
F2 is force that the output plunger applies to the car
A1 is Area of input piston
A2 is area of larger piston
We are given;
R2/R1 = 9
So,R2 = 9R1
F1 = 48N
Area of input piston;
A1 = π(R1)²
Area of output piston;
A2 = π(9R1)²
Since, (F1/A1) = (F2/A2)
Thus;
F1/(π(R1)²) = F2/(π(9R1)²)
If we simplify, π(R1)² will cancel out to give;
F1 = F2/9²
Thus;
F2 = 9² x F1
Plugging in 48N for F1, we have;
F2 = 9² x 48
F2 = 81 x 48
F2 = 3888N
Final answer:
Using the principle of Pascal's law and ignoring the height difference, the output force is found by the formula F2 = F1*(r2/r1)^2. Given F1 is 48N and r2/r1 is 9.0, the output force F2 equates to 3888N.
Explanation:
In the case of a hydraulic jack, the principle of Pascal's law is applied. According to this law, pressure applied at one point of the fluid is transmitted equally in all directions. Therefore, if we ignore the height difference between pistons, the pressure exerted on both pistons would be the same.
Pressure is equal to the force divided by the area, where area equals π times the radius squared (π*r^2). So, the pressure at the input piston (P1) is the force at the input piston (F1) divided by its area (A1): P1 = F1/A1, where A1 = π*(r1)^2.
For the output plunger(P2 = F2/A2), where F2 = force at the output plunger and A2 = π*(r2)^2. By equating the pressures (P1=P2) and simplifying, we find that F2 = F1*(r2/r1)^2, where r2/r1 is given as 9.0. So, the output force F2 would be 48N*(9.0)^2 = 3888N.
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