*4-12. Determine the angle 0 (0° < 0 < 180°) of the force F so that it produces a maximum moment and a minimum moment about point A. Also, what are the magnitudes of these maximum and minimum moments?
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- Write the equations that describe the simple harmonic motion of a particle moving uniformly around a circle of radius8units, with linear speed 3units per second.Let f(x)= 4xex - sin(5x). Find the third derivative of this function. Note ex is denoted as e^x below. Select one: (12+4x^3)e^x + 125sin(5x) 12e^x + 125cos(5x) not in the list (12+4x)e^x + 125cos(5x) (8+4x)e^x + 25sin(5x)Calculate the period of oscillation of ?(x,t) for a particle of mass 1.67 × 10-27 kg in the first excited state of a box of width 1.68 × 10-15 m. Include a sketch of U(x) and ?(x).
- Consider a bob attached to a vertical string. At t = 0, the bob is displaced to a new position where thestring makes −8° with the vertical and is then released from rest. If the angular displacement θ(t) iswritten as a cosine function, then the phase constant φ is:Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit τ → 0) for overdamping. Sketch the response functions.Let the initial position and speed of an overdamped, nondriven oscillator be x0 and v0, respectively. (a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values A1=2x0+v021 and A2=1x0+v021 where 1 = 2 and 2 = + 2. (b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by x=2x, otherwise the asymptotic paths are along the other dashed curve given by x=1x. Hint: Note that 2 1 and find the asymptotic paths when t .
- Q # 01: Consider a ball of 100g dropped with zero velocity from the height of 2m. Estimate its total energy in eV units. Using the energy conservation, find out the velocity as a function of position of the ball. Sketch its phase trajectory. Calculate the time it takes to reach the ground. Let’s assume that it is bounced back with no loss in its total energy.Will it reachthe same height? Make an analytical argument. What if the collision with the ground is not elastic and it loses some of its energy (which energy?).The ball will eventually come to rest after bouncing few times. Sketch the phase trajectory for the whole duration. What is the range of total energy of this system? Can the energy of this systemassume discrete values? Explain mathematically.An electron with mass 9.11 x 10^-31 kg is undergoing simple harmonic motion as it orbits around a Hydrogen nucleus with a frequency 7 x 10^15 Hz and amplitude of 5.3 x 10-11 m. Calculate the a) Period of the oscillationb) Angular velocityc) Maximum velocityd) Maximum accelerationd) Maximum force acting on the objecte) Maximum potential energy of the object (assume no energy loss)An electron undergoes simple harmonic motion with the acceleration shown below: ax(t)=−amaxsin(2t/T) with amax=5839 ms2 and T=316 seconds. Assuming that the only motion is oscillatory (ignoring overall translation), what is the maximum speed of the electron? What is the amplitude of the electron's position?
- Show,by integration that the moment of inertia of a uniform thin rod AB of mass 3m and length 4a about an axis through one end and perpendicular to the plane of the rod, is 16ma2 The rod is free to rotate in a vertical plane about a smooth horizontal axis through its end A Find i)The raduis of gyration of the rod about an axis through A ii)The period of small oscillations of the rod about its position of stable equilibrumConsider a bob attached to a vertical string. At t=0 the bob is displaced to a new position where the string makes -8 degree with tge vertical and is then released from rest.If the angular displacement is written as cosine function then the phase constant is?Explain the term amplitude as it relates to the graph of asinusoidal function.