17. Periodic Motion

At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine the maximum acceleration.

(III) A mass m is at rest on the end of a spring of spring constant k. At t = 0 it is given an impulse J by a hammer. Write the formula for the subsequent motion in terms of m, k, J, and t.

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 $0.500\(\operatorname{kg}\)$ mass on a spring has velocity as a function of time given by $v_{x}(t)=-(3.60\(\operatorname{cm}\)/s)\(\sin\)[(4.7\(\text{ }\)rad/s)t-(\(\pi\)/2)]$. What are the period and the force constant of the spring?

A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

(II) Figure 14–29 shows two examples of SHM, labeled A and B. For each write the equations for both A and B in the form of a sine or cosine.

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.090 m to x = -0.090 m?

Weighing Astronauts. This procedure has been used to 'weigh' astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

The wings of the blue-throated hummingbird (Lampornis clemenciae), which inhabits Mexico and the southwestern United States, beat at a rate of up to 900 times per minute. Calculate (a) the period of vibration of this bird's wings, (b) the frequency of the wings' vibration, and (c) the angular frequency of the bird's wing beats.

(II) A 0.25-kg mass at the end of a spring oscillates 3.2 times per second with an amplitude of 0.15 m. Determine the speed when it is 0.10 m from equilibrium.

The point of the needle of a sewing machine moves in SHM along the x-axis with a frequency of 2.5 Hz. At t = 0 its position and velocity components are +1.1 cm and -15 cm/s, respectively. Find the acceleration component of the needle at t = 0.

A $0.500\mathrm{kg}$ mass on a spring has velocity as a function of time given by $v_x\left(t\right)=-\left(3.60\mathrm{cm}/s\right)\sin \left[\right(4.7rad/s)t-(\pi /2\left)\right]$. What are the amplitude and the maximum acceleration of the mass?