A body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = (1.6m)*sin (πt + π/3). What is the velocity (in m/s) of the body at t = 1.3 s?
x = 1.6 (pi)cos(pi*1.3 +pi/3)
= 5.03 cos (1.63 pi)
= 5.03 * .397
To find the velocity of the body at t = 1.3 s, we need to take the derivative of the displacement equation with respect to time (t).
The given displacement equation is: x = (1.6 m) * sin(πt + π/3)
Let's differentiate it with respect to t:
dx/dt = d/dt [(1.6 m) * sin(πt + π/3)]
To differentiate the function sin(πt + π/3), we need to remember the chain rule.
Chain rule: d/dt [f(g(t))] = f'(g(t)) * g'(t)
Applying the chain rule to our equation:
dx/dt = (1.6 m) * d/dt [sin(πt + π/3)]
= (1.6 m) * cos(πt + π/3) * d/dt [πt + π/3]
= (1.6 m) * cos(πt + π/3) * (π)
Now, let's evaluate the equation at t = 1.3 s:
Velocity at t = 1.3 s, v(1.3) = (1.6 m) * cos(π(1.3) + π/3) * (π)
Calculating the value using a calculator:
v(1.3) ≈ -3.2 m/s
Therefore, the velocity of the body at t = 1.3 s is approximately -3.2 m/s.