BackTall buildings are designed to sway in the wind. In a 100 km/h wind, suppose the top of a 110-story building oscillates horizontally with an amplitude of 15 cm at its natural frequency, which corresponds to a period of 7.0 s. Assuming SHM, find the maximum horizontal velocity and acceleration experienced by an employee as she sits working at her desk located on the top floor. Compare the maximum acceleration (as a percentage) with the acceleration due to gravity.
A “seconds” pendulum has a period of exactly 2.000 s. That is, each one-way swing takes 1.000 s. What is the length of a seconds pendulum in Austin, Texas, where g = 9.793 m /s2? If the pendulum is moved to Paris, where g = 9.809 m/s2, by how many millimeters must we lengthen the pendulum? What is the length of a seconds pendulum on the Moon, where g = 1.62 m/s2?
(II) The human leg can be compared to a physical pendulum, with a “natural” swinging period at which walking is easiest. Consider the leg as two rods joined rigidly together at the knee; the axis for the leg is the hip joint. The length of each rod is about the same: assume 55 cm. Let the upper rod have a mass of 7.0 kg and the lower rod a mass of 4.0 kg. Calculate the natural swinging period of the system.
In Section 14–5, the oscillation of a simple pendulum (Fig. 14–48) is viewed as linear motion along the arc length 𝓍 and analyzed via F = ma. Alternatively, the pendulum’s movement can be regarded as rotational motion about its point of support and analyzed using T = Iα. Carry out this alternative analysis and show that θ (t) = θₘₐₓ cos (t + θ), where θ (t) is the angular displacement of the pendulum from the vertical at time t, as long as its maximum value is less than about .
The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?
Suppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance 𝓍 ≤ R from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius 𝓇 ≤ 𝓍 there is no net gravitational force from the mass in the spherical shell with 𝓇 > 𝓍. a. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of x, R, m, M, and any necessary constants.