- #1

vxr

- 25

- 2

- Homework Statement
- An air-track glider is attached to a spring pulled at distance ##d = 0.2m## to the right. Starting from released stage at ##t = 0## it subsequently makes ##n = 15## complete oscillations in time ##t = 10s##. Determine the period of oscillation ##T## and the object's maximum speed (velocity) ##v##, as well as its position and velocity ##v## at time ##t = 0.8s##.

- Relevant Equations
- ##x(t) = Acos(\omega t + \theta)##

So I am almost sure I know how to solve this, just curious about the maximum velocity. Anyway, if you could double check my calculations, here it is.

##T = \frac{t}{n} = \frac{10s}{15} = \frac{2}{3}s##

##\omega = \frac{2\pi}{T} = 2\pi \frac{3}{2} = 3\pi##

a). position at ##t = 0.8s##:

##x(t) = Acos{(\omega t + \theta)} = Acos\Big(\frac{2\pi t}{T} + \theta\Big) = Acos\Big( 3\pi t \Big)##

##x(0.8s) = 0.2 cos (7,54) = 0.061 m##

b). velocity at ##t = 0.8s##:

##v = \frac{d}{dt}x = \frac{d}{dt}\Big( Acos(\omega t + \theta) \Big) = -A\omega sin(\omega t + \theta)##

##v(0.8s) = -0.2 * 3\pi sin (3\pi * 0.8) = -\frac{3}{5} sin (\frac{12}{5}\pi) =~ -1.79 \frac{m}{s}##

c). maximum velocity. During my classes something like that was done:

##v_{max} = A\omega \cos (0) = A \omega = 0.2 * 9.42 = 1.884 \frac{m}{s}##

I understand that ##cos(0) = 1## but is this correct?

##T = \frac{t}{n} = \frac{10s}{15} = \frac{2}{3}s##

##\omega = \frac{2\pi}{T} = 2\pi \frac{3}{2} = 3\pi##

a). position at ##t = 0.8s##:

##x(t) = Acos{(\omega t + \theta)} = Acos\Big(\frac{2\pi t}{T} + \theta\Big) = Acos\Big( 3\pi t \Big)##

##x(0.8s) = 0.2 cos (7,54) = 0.061 m##

b). velocity at ##t = 0.8s##:

##v = \frac{d}{dt}x = \frac{d}{dt}\Big( Acos(\omega t + \theta) \Big) = -A\omega sin(\omega t + \theta)##

##v(0.8s) = -0.2 * 3\pi sin (3\pi * 0.8) = -\frac{3}{5} sin (\frac{12}{5}\pi) =~ -1.79 \frac{m}{s}##

c). maximum velocity. During my classes something like that was done:

##v_{max} = A\omega \cos (0) = A \omega = 0.2 * 9.42 = 1.884 \frac{m}{s}##

I understand that ##cos(0) = 1## but is this correct?