A particle of mass m is subject to an attractive central force k F(r) = For which values of the conserved quantities is the motion of the particle bounded?
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- Use the principle of minimum potential energy developed in Section 3.10 to solve the bar problems shown in Figure P3-52. That is, plot the total potential energy for variations in the displacement of the free end of the bar to determine the minimum potential energy. Observe that the displacement that yields the minimum potential energy also yields the stable equilibrium position. Use displacement increments of O.002 in., beginning with x = -0.004. Let E = 30 x 106 psi and A = 2 in2 for the bars. 10,000 Ib 30 in. 10,000 lb 50 in. (b) Figure P3-52In which situation's would angular acceleration be negative? Select all that apply 1. An object is at rest and is starting to rotate clockwise 2. An object is rotating clockwise and speeding up 3. An object is rotating clockwise and speeding up 4. An object is rotating counterclockwise and slowing downYou are working with a movie director and investigating a scene with a cowboy sliding off a tree limb and falling onto the saddle of a moving horse. The distance of the fall is several meters, and the calculation shows a high probability of injury to the cowboy from the stunt. Let's look at a simpler situation. Suppose the director asks you to have the cowboy step off a platform 2.55 m off the ground and land on his feet on the ground. The cowboy keeps his legs straight as he falls, but then bends at the knees as soon as he touches the ground. This allows the center of mass of his body to move through a distance of 0.670 m before his body comes to rest. (Center of mass will be formally defined in Linear Momentum and Collisions.) You assume this motion to be under constant acceleration of the center of mass of his body. To assess the degree of danger to the cowboy in this stunt, you wish to calculate the average force upward on his body from the ground, as a multiple of the cowboy's…
- The scalar triple product of three vectors is a • (b x c). Prove that the scalar triple product will not change when you cyclically permute the three vectors. (i.e., prove that a • (b x c) = b • (c x a) = c • (a x b) )Learning Goal: To understand the Equipartition Theorem and its implications for the mechanical motion of small objects. In statistical mechanics, thermal energy is the random motion of the microscopic world. The average kinetic or potential energy of each degree of freedom of the microscopic world therefore depends on the temperature. If heat is added, molecules increase their translational and rotational speeds, and the atoms constituting the molecules vibrate with larger amplitude about their equilibrium positions. It is a fact of nature that the energy of each degree of freedom is determined solely by the temperature. The Equipartition Theorem states this quantitatively: The average energy associated with each degree of freedom in a system at absolute temperature T is (1/2)k³T, where KB = : 1.38 × 10-2³ J/K is Boltzmann's constant. A "degree of freedom" corresponds to any dynamical variable that appears quadratically in the energy. For instance, (1/2)mv² is the kinetic energy of a…a.) A particle with mass m is dropped with zero initial velocity a height h above the ground, under the action of the gravitational field (acceration constant g), such that it reaches the ground (z=0) with zero potential energy and velocity v>0. Determine h, in terms of the other quantities given. b.) Another particle moves with Simple Harmonic Motion with centre O.The particle has velocity 13ms–1 when it is 3m from O and 5ms–1 when it is 5m from O.(i) Find the period and amplitude of the motion
- is the vector foeld conservative? prove it.Consider a roller-coaster car on a track that has a loop of known radius R. If there is no friction between the rollercoaster car and the track, then determine (a) the minimum speed at the top of the loop for the rollercoaster car to still be in contact with the track and (b) the minimum height h the rollercoaster car must start from in order to go all the way around the loop without losing contact with the track (assuming the rollercoaster starts from rest). The only knowns here are R and g, so your symbolic answers need to be in terms of these. Part (a) can be treated using newton's second law and uniform circular motion (even though it's not really uniform). Start with a free-body diagram of a coaster car at the top of the loop. What does it mean for the rollercoaster car to still be "in contact" with the track? Part (b) can be treated using conservation of mechanical energy. You'll need your result from part (a). How high above the ground are you when you're at the top…Consider the 3-dimensional force field ⃗ F = (x^2 − ze^y)⃗i + (y^3 − xze^y)⃗j + (z^4 − xe^y)⃗k:(a) Show that ⃗ F is conservative.(b) If ⃗ F is conservative, find the corresponding potential function f (x, y).(c) If an object travels on a path ⃗r (t), (t_0 < t < t_f ), does the work done bythe force field depend on the path taken? Find the work done on the object movingon the path ⃗r (t) by the force field ⃗ F , if ⃗r (t_0) = (3, 1, 2) and ⃗r (t_f ) = (6, 2, 5)