Concept explainers
A spacecraft of mass m describes a circular orbit of radius r1 around the earth. (a) Show that the additional energy ΔE that must be imparted to the spacecraft to transfer it to a circular orbit of larger radius r2 is
where M is the mass of the earth. (b) Further show that if the transfer from one circular orbit to the other is executed by placing the spacecraft on a transitional semielliptic path AB, the amounts of energy ΔEA and ΔEB which must be imparted at A and B are, respectively, proportional to r2 and r1:
Fig. P13.116
(a)
Show that additional energy
Answer to Problem 13.116P
The additional energy
Explanation of Solution
Given information:
The minimum distance between the center of the earth to the point A is
The maximum distance between the center of the earth to the point B is
The mass of the earth is M.
Calculation:
Show the figure with the force acting as in Figure (1).
The expression for the normal acceleration
The expression for calculating the geocentric force acting on the spacecraft when it is on the surface of earth (F) as follows:
Here, G is the universal gravitational constant and M is the mass of the earth.
Calculate the velocity of the circular orbit (v) by considering the force equilibrium by taking Newton’s second law using the relation:
Substitute
The expression for the kinetic energy in the circular orbit (T) as follows;
The expression for the potential energy in the circular orbit (V) as follows;
Calculate the energy required (E) for the spacecraft using the relation:
Substitute
Substitute
The expression for the energy required for the circular orbit of radius
The expression for the energy required for the circular orbit of radius
Calculate the addition energy imparted to the spacecraft to transfer it to circular orbit
Substitute
Therefore, the additional energy
(b)
Show the transfer from one circular orbit to the other is executed by placing the spacecraft on transitional semi elliptic path AB, the amounts of energy
Answer to Problem 13.116P
The amount of energy imparted at A
Explanation of Solution
Given information:
The minimum distance between the center of the earth to the point A is
The maximum distance between the center of the earth to the point B is
The mass of the earth is M.
Calculation:
Consider the circular orbit of radius
The expression for the velocity of the circular orbit
Calculate the kinetic energy at the circular orbit
Substitute
Consider that the after the spacecraft engines are fired and it is placed on a semi-elliptic path AB.
The expression or the principle of conservation of angular momentum at point A to the point B as follows:
The expression for the kinetic energy at point B
Here, m is the mass of the satellite.
The expression for the gravitational potential energy at point B
The expression for the kinetic energy of the orbit at point A
The expression for the gravitational potential energy at point A
The expression for the principle of conservation of energy at the point A to point P as follows:
Substitute
Substitute
Simplify the Equation,
Substitute
Calculate the kinetic energy in the semi elliptic path AB
Substitute
Calculate the additional energy
Substitute
Divide and Multiply by
Substitute
Calculate the kinetic energy in the semi elliptic path AB
Substitute
Calculate the additional energy
Substitute
Divide and Multiply by
Substitute
Therefore, the amount of energy imparted at A
Want to see more full solutions like this?
Chapter 13 Solutions
VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)
- No chatgpt plsarrow_forwardA six cylinder petrol engine has a compression ratio of 5:1. The clearance volume of each cylinder is 110CC. It operates on the four-stroke constant volume cycle and the indicated efficiency ratio referred to air standard efficiency is 0.56. At the speed of 2400 rpm. 44000KJ/kg. Determine the consumes 10kg of fuel per hour. The calorific value of fuel average indicated mean effective pressure.arrow_forwardThe members of a truss are connected to the gusset plate as shown in (Figure 1). The forces are concurrent at point O. Take = 90° and T₁ = 7.5 kN. Part A Determine the magnitude of F for equilibrium. Express your answer to three significant figures and include the appropriate units. F= 7.03 Submit ? kN Previous Answers Request Answer × Incorrect; Try Again; 21 attempts remaining ▾ Part B Determine the magnitude of T2 for equilibrium. Express your answer to three significant figures and include the appropriate units. Figure T₂ = 7.03 C T2 |? KN Submit Previous Answers Request Answer × Incorrect; Try Again; 23 attempts remaining Provide Feedbackarrow_forward
- Consider the following acid-base reaction: Fe3+(aq) +3H2O -Fe(OH)3 (s) + 3H* ← A. Using thermodynamics, calculate the equilibrium constant K at 25°C (The AG° of formation of Fe(OH)3(s) is -699 kJ/mol). B. Using the value of K you calculated in part a, if a solution contains 10-4 M Fe3+ and has a pH of 7.5, will Fe(OH)3(s) precipitate? Show all calculations necessary to justify your answer. Note that the reaction as written is for precipitation, not dissolution like Ksp-arrow_forwardA vertical force of F = 3.4 kN is applied to the hook at A as shown in. Set d = 1 m. Part A 3 m 3m 0.75 m 1.5 m. Determine the tension in cable AB for equilibrium. Express your answer to three significant figures and include the appropriate units. FAB= Value Submit Request Answer Part B Units ? Determine the tension in cable AC for equilibrium. Express your answer to three significant figures and include the appropriate units. FAC = Value Submit Request Answer Part C ? Units Determine the tension in cable AD for equilibrium. Express your answer to three significant figures and include the appropriate units.arrow_forwardConsider the heat engine operating at steady state between the two thermal reservoirs shown at the right while producing a net power output of 700 kW. If 1000 kW of heat (Q̇H) is transferred to the heat engine from a thermal reservoir at a temperature of TH = 900 K, and heat is rejected to a thermal reservoir at a temperature of TL = 300 K, is this heat engine possible? Can you answer this question for me and show all of the workarrow_forward
- 1.12 A disk of constant radius r is attached to a telescoping rod that is extending at a constant rate as shown in Fig. P1.12. Both the disk and the rod are rotating at a constant rate. Find the inertial velocity and acceleration of point P at the rim of the disk. ท2 L 0 SS P α e 0 O' êL Fig. P1.12 Rotating disk attached to telescoping rod. 60 LLarrow_forwardTwo different options A and B with brake pads for disc brakes are connected to the rope drum. The diameter of the rope drum is 150 mm. What distance must the pads B be at from the center of rotation to cover the same distance as A?A B- Width 50 mm - Width 60 mm- Evidence center 120mm - Construction power 900 N from rotation center.- Maintains a weight of 200 kgwhen the installation force is 1.4kN (μ is missing from the data)M=μF(Ry-Ri)Right answer R=187 mmarrow_forwardAssume the xy plane is level ground, and that the vertical pole shown in the diagram lies along the z-axis with its base at the origin. If the pole is 5 m tall, and a rope is used to pull on the top of the pole with a force of 400 N as shown, determine the magnitudes of the parallel and perpendicular components of the force vector with respect to the axis of the post i.e. with respect to the z-axis.arrow_forward
- 4-1 Q4: Q5: (20 Marks) Find √48 using False Position Method with three iterations. Hint: the root lies between 3 and 4. (20 Marks)arrow_forwardDetermine the angle between vectors FA and FB that is less than 180 degrees. FA is the vector drawn from the origin to point A (-4, 4, 2) while FB is the vector drawn from the origin to point B (3, 1, -3).arrow_forwardFind the resultant force vector from adding F1, F2 and F3, where … F1 = {-8i+10j-32k} N F2 is 40 N in magnitude with coordinate direction angles α, β, and γ, of 45, 120 and 60 degrees, respectively and F3 is 22 N in magnitude with transverse and azimuth angles of 65 and 40 degrees, respectively Express your final answer as a Cartesian vector as well as a magnitude with angles.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY