the mass or the carth is Me, show thnat the gravitational potential energy of a body of mass m located a distance r from the center of the earth is V, = -GM¸m/r. Recall that the gravitational force acting between the earth and the body is F = G(M¸m/P), Eq. 13-1. For the calculation, locate the datum at r→ ∞. Also, prove that F is a conservative force.
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- Prove the following equation Pw(x,t)= py(x,t) with p=2 ww in (2x-1) Knowing that y (x,t)= 3e^Consider an object of mass m moving in a horizontal circle of radius r on a rough table. It is attached to a string fixed at the center of the circle. The speed of the object is initially vo. After completing one full trip around the circle, the speed is ½ vo. (a) Find the energy dissipated by friction during that one revolution in terms of m and vo. (b) What is the coefficient of kinetic friction in terms of g, r and vo? c) How many more revolutions will the object make before coming to rest?The potential energy of a conservative force is U(x) = ax?-bx", with a and b positive constant parameters. (a) Find the equilibrium points and discuss whether these points are stable or unstable (and why). (b) At time t=0, a particle with mass m and initial velocity Vin=0 is placed at a distance xin close to the stable equilibrium point. Assume that Xin is very small compared with the distance between the stable and the unstable equilibrium points. Find an approximate expression for the period of oscillations of the mass m. (c) A second particle with mass M is placed, at time t=0, at the exact position of the stable equilibrium point, with a non-zero initial velocity vin. Find the minimum value of vin such that the particle can travel a distance at least equal to the distance between the stable equilibrium point and the nearest unstable equilibrium point. This problem must be solved symbolically, numbers are not required.
- Consider a mass, m, moving under the influence of an effective potential energy b 7.2 U(r) = r where a and b are positive constants and r is the radial distance from the origin. In this case, U(r) is a 1D potential energy. (a) Generate a simple plot the potential energy, U(r) vs r. (b) Next, find the equilibrium distance, ro, for the mass in this potential. Then evaluate minimum potential energy U(r.).Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k >0. We know physically that our spring will eventually come to rest nomatter the initial conditions or the values of m, β, or k. If our modelis a good model, all solutions x(t) should approach 0 as t → ∞. Foreach of the three cases below, explain how we know that both rootsr1,2 =−β ± Sqrt(β^2 − 4km)/2mwill lead to solutions that exhibit exponentialdecay.(a) β^2 − 4km > 0. (b) β^2 − 4km =0. (c) β^2 − 4km >= 0.(d) The force, F, of a turbine generator is a function of density p, area A and velocity v. By assuming F = apªA® v° and dimensional homogeneity, find a, b and c and express F in terms of p, A and v. (a, a, b and c are real numbers). Make the following assumptions to determine the dimensionless parameter: F = 1 k N if the scalar values of pAv= 1milli. (e) The dynamic coefficient of viscosity µ (viscosity of a fluid) is found from the formula: µAv F =
- Task 1 (a) A low voltage transformer manufacturing line has a purchase order of 1700 piece. If the manufacturing line production rate is 22 piece per minute use dimensional analysis techniques to determine the time taken to produce the requested transformers to the nearest minute in hours and minutes. (b) Assume that the production time of a transformer can be given by the following formula: 1 t = Gmav-312 Where m is the mass of the transformer in kg, v is the velocity of the manufacturing line in m/s and l is the length of the production line in meters. Find the dimensions of G. (c) In b, if m = 220 grams, 1= 10.2 m, v=2.3 m/s and time is 30.1 ms. Find G?A particle of mass moves in 1 dimensional space with the following potential energy: (in figure) where U0 and alpha are positive, 1) What is the minimum velocity a particle must have to go from the origin to infinity? 2) What is the angular frequency w of the oscillations around the stable equilibrium point?In a clamped frictionless pipe elbow (radius R) glides a sphere (weight W = mg) with zero initial velocity downwards from the top. %3D Determine the support reactions at the Clamping (wall connection) in dependence on the position o of the sphere. At which o the reactions take extreme values?
- A 200 g mass is attached to a spring, just like in activity 17c. The mass is lifted up 5 cm and released so that it begins to oscillate about the equilibrium point. The spring has a spring constant of k=500 N/m (500 J/m2). b) On the same graph, quickly sketch (without calculating values) the PEspring-mass, Etotal, and KE of the system if the mass were initially pulled back (Stretched) 2.5 cm from its equilibrium point, instead of lifted up (compressed) 5 cm.Question a) A satellite of mass m = 3.6kg orbits the Earth 300km ab ove the Earth'ssurface. How much do es the p otential energy of the satellite change when it islaunched from the surface of the Earth to its orbit? b) By assuming the orbit is circular and the satellite is held in its orbit bygravity, show that the p erio d T of the satellite's orbit can b e expressed as(see image)where r is the distance of the satellite from the Earth's centre and g is theacceleration due to gravity. c) Using the mean radius of the Earth of R = 6400km, calculate the tangentialspeed of the satellite in its orbit. d) If the satellite is launched by ro cket from the Equator, how much do es thekinetic energy of the satellite change when it is placed into this orbit? Do esmost of the energy supplied by the ro cket to the satellite go into the satellite'skinetic energy or the p otential energy?Consider a mass, m, moving under the influence of an effective potential energy -a b T 7-2 U(r) = + " where a and b are positive constants and r is the radial distance from the origin. In this case, U(r) is a 1D potential energy. (c) Expand The function for U(r) for small displacements about the equilibrium point, To. This will be a Taylor series for U(r) in terms of (r-ro)", where n is an integer. Generate the first three nonzero terms in the series.