17. Periodic Motion

The displacement of an oscillating object as a function of time is shown in Fig. E14.4. What is (c) the period? (d) the angular frequency of this motion?

The displacement of an oscillating object as a function of time is shown in Fig. E14.4. What is (a) the frequency? (b) the amplitude?

(II) Consider two objects, A and B, both undergoing SHM, but with different frequencies, as described by the equations x_A = (2.0 m) sin (4.0 t) and x_B = (5.0 m) sin (3.0 t), where t is in seconds. After t = 0, find the next three times t at which both objects simultaneously pass through the origin.

Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in FIGURE P15.46. What is her speed when the spring's length is 1.2 m?

A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 c/m when 𝓍 = -5.0 cm What is the block's position when the acceleration is maximum?

It has recently become possible to 'weigh' DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.58 shows a thin rectangular cantilever etched out of silicon (density 2300 kg/m³) with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight—but measurable—decrease in the oscillation frequency. A vibrating cantilever of mass M can be modeled as a block of mass ⅓M attached to a spring. (The factor of ⅓ arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately—perhaps to only two significant figures—but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. What was the mass of the DNA?

A mass hanging from a spring oscillates with a period of 0.35 s. Suppose the mass and spring are swung in a horizontal circle, with the free end of the spring at the pivot. What rotation frequency, in rpm, will cause the spring's length to stretch by 15%?

A 1.0 kg block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40 cm/s. What are The block's speed at the point where 𝓍 = (½)A?

The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t - π/4), where t is in s. Determine: The velocity at t = 0.40 s.

The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t - π/4), where t is in s. Determine: The initial conditions.

A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has v_x = -30 cm/s. Determine: The position at t = 0.40 s.

A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has v_x = -30 cm/s. Determine: The maximum speed.

A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if the amplitude is doubled?

A 1.00 kg block is attached to a horizontal spring with spring constant 2500 N/m. The block is at rest on a frictionless surface. A 10 g bullet is fired into the block, in the face opposite the spring, and sticks. What was the bullet's speed if the subsequent oscillations have an amplitude of 10.0 cm?

A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.