1. A particle executes simple harmonic motion with an amplitude of 10 cm and time period 6 s. At t = 0 it is at position x = 5 cm going towards positive x-direction. Write the equation for the displacement x at time t. Find the magnitude of the acceleration of the particle at t = 4 s. %3D
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- C. Why was h measured to the center of the bob? d. Why was the completed number of oscillation chosen to be large? e. Can you see any advantage in measuring the height of the ceiling in this way?2. Consider a particle undergoing simple harmonic motion. The velocity of the particle at position x¹ is v¹ and velocity of the particle at position ² is ². Show that the ratio of time period and amplitude is: T A = 2t. √v²x²-v²x²A mass m = 3.3 kg is at the end of a horizontal spring on a frictionless horizontal surface. The mass is oscillating with an amplitude A = 4.5 cm and a frequency f = 1.5 Hz. a. Write an equation for the spring constant k. b. Calculate the spring constant k, in Newtons per meter. c. Write an equation for the total mechanical energy, E, of the motion. Your expression should be in terms of the variables in the original problem statement. d. Calculate the total mechanical energy E, in joules.
- A particle moves with simple harmonic motion in a straight line. At time t = 0, the acceleration is 3 m/s2 , the velocity is 1 m/s and the displacement is -0.30 m. Find the period of motion, in seconds.The airplane on the amusement park ride moves along a path defined by the equations r= 4 m, 0 = (0.2t) rad, and z = (0.5cose) m, where t is in seconds. Determine the transverse component of the velocity when t = 6 s. %3D r = 4 mA Ferris wheel has a radius of 25 m and its centre is 27 m above the ground. It rotates once every 40 seconds. Sandy gets on the Ferris wheel at its lowest point and then the wheel starts to rotate. a) Determine a sinusoidal equation that gives Sandy's height, h, above the ground as a function of the elapsed time, t, where h is in metres and t is in seconds. b) Determinc the first time, t (in scconds), when Sandy will be 35 m abovc the ground.
- Q1. Consider a spring mass system given in Fig 1 with mass of 1kg attached to a spring with K=10. The motion is damped with C=6 and is being driven by a force given by 25 cos 4t . If the spring is released from rest, find the equation of displacement y(t). Note: You cannot use the direct formulas for constants in particular solution. Spring Mass r(t) Dashpot Fig. 1Hz A cart of mass 200.0 g is attached to a horizontal spring of spring constant 75 N/m. The cart has four wheels; each wheel is a solid disk of radius 1.3 cm and mass 50.0 g. The cart is free to roll with no internal friction on a horizontal surface. Including the rotational inertia of the wheels, what is the frequency at which the cart will oscillate back and forth?1. A 3.5 kg steel ball is swung at a constant speed in a vertical circle of radius 1.2 m, on the end of a light, rigid steel rod, as illustrated. If the ball has a frequency of 1.0 Hz, calculate the tension in the rod due to the mass at the top (A) and at the bottom (B) positions.
- A yo-yo oscillates up and down like a sine function y(t)=sin(t). a) What is the displacement over the time interval [0,6π]? b) What is the total distance traveled by the yo-yo over the same time interval? c) What is the minimum velocity of the yo-yo? Minimum speed? d) At what part of the cycle is the yo-yo increasing in speed but decreasing in velocity?RjA block of mass M is attached to two springs with spring constant K1 and K2, respectively, as shown in the figure. When the block is in the initial position, both springs are in their rest position. The block is dragged to the right a distance X, and then released. Find: 1. An expression for the horizontal acceleration of the block as a function of M, K1, K2, X. 2. An expression for the period of the oscillation of the block as a function of M, K1, K2. 3. An expression for the magnitude of the velocity of the block as it passes through the initial point again, as a function of M, K1, K2, X.