A door stopper spring is flicked from its equilibrium position, and the angular velocity of the plastic end is given by the function: @ (t) = 5e-0.21 sin(4t) 1. Find its angular acceleration function a (t). 2. Find its angular displacement function 0 (t), assuming 0 (0) = 0. 3. If the rotational motion is confined to the xy plane, what is the direction of the acceleration at t=0? Justify your answers with your rationale and equations used.
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- Consider a “round” rigid body with moment of inertia I = BMR2, where M is the body’s mass, R is the body’s radius, and B is a constant depending on the type of the body. The center of the “round” rigid body is attached to a spring of force constant k, and then the body is made to roll without slipping on a rough horizontal surface. Due to the spring, it is expected the body will oscillate by rolling back and forth from its resting position. A. Determine the angular frequency and the period for small oscillations of the round rigid body. Express your answer in terms of B.A 1.80 kg monkey wrench is pivoted 0.250 m from its center of mass and allowed to swing as a physical pendulum. The period for small-angle oscillations is 0.940 s. (a) What is the moment of inertia of the wrench about an axis through the pivot? (b) If the wrench is initially displaced 0.400 rad from its equilibrium position, what is the angular speed of the wrench as it passes through the equilibrium position?A solid cylinder of mass M and radius R is mounted to an axle through its center. The axle is attached to a horizontal spring of constant k, as shown in the figure. Initially the cylinder is at rest and the spring is un-stretched. The cylinder is then pulled a distance A and released. The cylinder rolls back and forth without slipping. A) Determine the angular frequency ωfreq and period T of the simple harmonic motion of this spring-rolling mass system? Express your answer in terms of k and M. B) Assume that we have the same spring and mass system, onlynow the cylinder is released from rest on a frictionless surface at a distance A from the equilibrium position. What is the period of the simple harmonic motion of this system? Express your answer in terms of k and M.
- 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 equilibrumA child with mass of 10 kg gets into a toy car with mass of 65 kg on a playground, causing it to sink on its spring (withspring constant 327 N/m). An adult walks by and gives the top of the car a shove, causing it to undergo oscillationswith amplitude 30 cm in the vertical direction. Assuming the oscillations are simple harmonic, what is the angularfrequency of the oscillations?ω = (in rad/s) a. 0.673 b. 1.715 c. 2.088 d. 2.243 e. 5.717mass m = 0.16 kg attached to a thread of length L = 1.5 m The pendulum is released from the horizontal position A as shown in the figure. The angular velocity of the mass as it passes through point B, where β = 120 °, is:
- The angular moomentum of a point with a mass of m, a velocity v, and position r is: L = mr x v. Prove that the rate of change of L, dL/dt is equal to the torque (T = r x F). F is the force on the point.The ball as shown moves in a circle of radius 0.50 m. At t = 0, the ball is located on the left side of the turntable, exactly opposite its position as shown. What are the correct values for the amplitude and phase constant (relative to an x axis to the right) of the simple harmonic motion of theshadow? (a) 0.50 m and 0 (b) 1.00 m and 0 (c) 0.50 m and π (d) 1.00 m and πA simple pendulum with mass m = 1.2 kg and length L = 2.48 m hangs from the ceiling. It is pulled back to an small angle of θ = 11° from the vertical and released at t = 0. a) What is the angular displacement at t = 3.54 s? (give the answer as a negative angle if the angle is to the left of the vertical) b) What is the magnitude of the radial acceleration as the pendulum passes through the equilibrium position?