ENGIN 52 HW 6 A particle of mass m is projected from Point A with an initial velocity v, perpendicular to line OA and moves under a central force F along a semicircular path of diameter OA. Observing that r=r, cose and using Eq. (12.25), show that the speed of the particle is v=v,/cos 0. TO
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- Problem 9.9 Consider a planet of mass M and radius R. Assume the planet is spherical and has a constant density. By direct integration determine the gravitational field and the gravitational potential at all points inside and outside the planet. (Assume the potential is zero at infinity and there are no other bodies in the universe.)Tarzan, mass 71 kg, holds on to a vine of length 10 m. He is initially at rest and the vine is initially horizontal. Tarzan then uses the vine to swing, going in a circular arc. Calculate the tension in the vine, in N, when the vine has become vertical. (Please answer to the fourth decimal place - i.e 14.3225)We have two spheres (m and M) that are separated by a small distance; m is to the left of M. A sphere of mass m (identical to one of the two initial spheres) is moving towards m at a speed V0. Show that when M is smaller than or equal to m, there will be 2 collisions and calculate the final speeds. Show that when M is larger than m, there will be three collisions and calculate the final velocities.
- A block A, of mass m = 10 Kg, compresses a spring of constant K = 1000 N / m in a length x = 3 cm. Starting from rest, the block is released, which moves from that moment on a horizontal surface without friction until it collides with another block B of mass m = 40 Kg, which was at rest. (Perfectly inelastic shock) and together they go up the channel (inclined surface) without friction, to later continue along a second horizontal plane without friction, at a height h with respect to the first (see Figure). Determine the energy variation that occurs in the collision .Consider two particles: p at the origin (0,0,0) = R³ with mass M > 0, and q at the point/position vector 7 = (x, y, z) = R³ with mass m > 0. Let G be the universal gravitational constant. (We will assume the MKS system of units.) The force F = F (7) felt by the particle q due to its gravitational interaction with particle p is: GMm 7(7)= == 7, for all 7 = (x, y, z) € R³\{0} . 17 Also consider the function ƒ : R³\{♂} → R given by GMm f(x, y, z) := TT , for all 7 = (x, y, z) € R³\{0} . Fix an arbitrary point/position vector = (x, y, z) in R³\{♂}. 2, calculate the (3) Calculat cade of the vector (4) Calculate the direction of the vector ₹(7). (5) Assume that is the total force on the particle q. Calculate the instantaneous acceleration, d, of the particle q when it is at the point 7 = (x, y, z).A 12 kg space rock with velocity vector v1 = 7 i (m/s) collides with and sticks to an 22 kg space rock with velocity vector v2 = 21 j (m/s). Calculate the final speed of the pair, in m/s. (Please answer to the fourth decimal place - i.e 14.3225)
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