Newton's Law of Gravitation2. The magnitude of the acceleration of an object under the pull of Earth's gravity isgiven by Newton's Universal Law of GravitationМЕa = GR?where G is the universal gravitational constant, ME is the mass of Earth, and R is thedistance of the object from the center of Earth.Let x be the distance above Earth's surface. We can rewrite the formula for theacceleration as a function of x by noting that R = Rp + x, where Rp is the radius ofEarth. Therefore,МЕa(x) = G-(RE + x)2d.(a) Show thatdx11(1 – x)*- x.(b) Use the above fact, along with the power series of1to determine a power1- x1series for(1+x)²*(c) What is the radius of convergence for the series in part (b)? (Hint: You do notneed to calculate anything. What is the radius of convergence for the power seriesof1does not change the radius of convergence.)-? This series has the same radius of convergence since taking a derivative (d) Rewrite a(x) asGME1a(x)R (1+x/RE)?and use the information from part (b) to determine a power series for a(x).(e) What is the radius of convergence, in terms of x, for the power series you foundin part (d)? (Hint: Use part (c).)(f) The constants have the values G = 6.674 × 10-11 m³kg¬'s-2, ME = 5.972 × 1024kg, and RE = 6.371 × 106 m. Substitute these values into the first term of theseries you found in part. This is the zeroth degree Taylor polynomial, a T,(x).The resulting value might be familiar. What is this value?

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a) 0) State Newton's law of gravitation. Give the meaning of any symbol you use. (1) Define gravitational field strength. (iii) Use your answers to (i) and (ii) to show that the iaagnitude of the gravitational field strength at the Earth's surface is GM R where Mis the mass of the Earth, R is the radius of the Earth and G is the gravitational constant.
Use the value of g = 9.80 [m/s^2]
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1. Gravity: a. What is the magnitude and direction of the gravitational force that the Earth exerts on you in Sl units? Make a reasonable numerical estimate and include units. b. The SI unit of force is a newton. What is a newton in terms of the more basic units, kilograms (kg), meters (m), and seconds (sec)? What is the magnitude and direction of the gravitational force that you exert on the Earth? Consider your answer to part a. 2. Contact Forces¹: Besides gravity, nearly all other forces we will encounter in this course are contact forces, which means that the two objects must be touching for the objects to exert forces on each other. Normal Force: When a surface is touching an object, the surface exerts a force on the object that is perpendicular to the surface and pushes into the object. This is called a "normal" force (normal means perpendicular in math). The harder the surfaces press against each other, the larger the normal force.²
2. Integrate the equations of motion for a spherical pendulum, i.e., a particle of mass m moving on the surface of a sphere of radius / in a gravitational field, with Lagrangian L = ml² -(0² + psin² 0) + mglcos 0. 2 Hint: Merely set up the integrals, do not evaluate them. You should obtain two integrals w.r.t. de: one determining t and another determining p. Express the first integral in terms of E,Ueff, m, and l. Express the second integral in terms of E,U eff, m, l, and M₂.
Hiker Harry walks 4 km east then 7 km 25o north of west. What is the magnitude and direction of Harry’s position after his hike as measured from his starting point? Give the angle measured relative to east. For example 90° means directly north.
B 3. Given the lengths of vectors A, B, and C, and the angle between A and B. Notice that A = B + С. Prove the law of cosines: |C|² = |Ã₁² + |B|² - 2|A||B| Cos 0. Then, show that if A and B are perpendicular, the law of cosines reduces to the familiar Pythagorean theorem.
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According to Lunar Laser Ranging experiments the average distance LM from the Earth to the Moon is approximately 3.85 × 10° km. The Moon orbits the Earth and completes one revolution in approximately 27.5 days (a sidereal month). Select the correct expression to calculate the velocity of the Moon. Select one: L'M O a. UM 2nLM O b. UM Тм O c. UM LM Тм 2nLM TM O d. UM
The chaos of the pandemic is finally over, and you decide to drive to your friend's house to pick her up, and take her back to your house before going out. She lives 4 miles straight East from you. a. What is the displacement of your journey in miles? b. What is the distance of your journey in miles?
1. Draw and label the vector with these components. Then determine the magnitu angle of the vector. a- A, = 3 ,Ay = -2 b- By = -2 ,By = 2 2. What is the speed of horse in meter per second that runs a distance of 1.2 miles- minutes? 3. A car moving along X- axis according on the function: X (t) = 2 t² + 5 t +7 on [1,2] Find: a- The average velocity. b- The acceleration at 3s. 4. Julie is walking around a track at a 2m/s for some exercise. She then decides t gging so she accelerates at a rate of 0.5m/s² for 3 seconds. ow far did Julie travel from the time she started to accelerate to the end of the 3 seco 5. Ralph uses a 2000 N force to push his car along a road for 1000 m. It takes hin to do this. a- Calculate the amount of work that Ralph did on the car. b- Calculate the amount of power that he generated.

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