Problem 13
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.
Problem 15
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. (a) Find the acceleration component of the needle at \(t =\) 0. (b) Write equations giving the position, velocity, and acceleration components of the point as a function of time.
Problem 16
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 (a) from \(x =\) 0.180 m to \(x = -\)0.180 m and (b) from \(x =\) 0.090 m to \(x = -\)0.090 m?
Problem 17
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?
Problem 20
A 0.500-kg mass on a spring has velocity as a function of time given by \({v_x}(t) = -\)(3.60 cm/s) sin[ (4.71 rad/s)\(t - \pi\)/2) ]. What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?
Problem 21
A 1.50-kg mass on a spring has displacement as a function of time given by $$x(t) = 7.40 \mathrm{cm}) \mathrm{cos} [ (4.16 \mathrm{rad}/s)t - 2.42] $$ Find (a) the time for one complete vibration; (b) the force constant of the spring; (c) the maximum speed of the mass; (d) the maximum force on the mass; (e) the position, speed, and acceleration of the mass at \(t =\) 1.00 s; (f) the force on the mass at that time.
Problem 22
In February 2004, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just 30 nm long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached \((f_S+\,_V)\) to the frequency without the virus \((f_S)\) is given by \(f_S+\,_V/f_S = 1\sqrt{ 1 + (m_V/m_S) }\), where \(m_V\) is the mass of the virus and \(m_S\) is the mass of the silicon sliver. Notice that it is \(not\) necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of 2.10 \(\times\) 10\(^{-16}\) g and a frequency of 2.00 \(\times\) 10\(^{15}\) Hz without the virus and 2.87 \(\times\) 10\(^{14}\) Hz with the virus. What is the mass of the virus, in grams and in femtograms?
Problem 23
A guitar string vibrates at a frequency of 440 Hz. A point at its center moves in SHM with an amplitude of 3.0 mm and a phase angle of zero. (a) Write an equation for the position of the center of the string as a function of time. (b) What are the maximum values of the magnitudes of the velocity and acceleration of the center of the string? (c) The derivative of the acceleration with respect to time is a quantity called the \(jerk\). Write an equation for the jerk of the center of the string as a function of time, and find the maximum value of the magnitude of the jerk.
Problem 25
A small block is attached to an ideal spring and is moving in SHM on a horizontal frictionless surface. The amplitude of the motion is 0.165 m. The maximum speed of the block is 3.90 m/s. What is the maximum magnitude of the acceleration of the block?
Problem 26
A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. The amplitude of the motion is 0.250 m and the period is 3.20 s. What are the speed and acceleration of the block when \(x =\) 0.160 m?
