A weight oscillates in a vertical motion according to the position function y(t)=-5 cos(t). Assuming t≥0, when will the acceleration if the weight be zero for the first time?
y" = 5cos(t)
so, when is that zero?
when t=90?
best use radians, pi/2
To find the time when the acceleration of the weight is zero for the first time, we need to differentiate the position function twice with respect to time.
Given:
Position function: y(t) = -5cos(t)
First derivative with respect to time:
v(t) = dy(t)/dt
= d/dt (-5cos(t))
= 5sin(t)
Second derivative with respect to time:
a(t) = dv(t)/dt
= d/dt (5sin(t))
= 5cos(t)
Now, we set the second derivative equal to zero and solve for t:
5cos(t) = 0
Since we want the first time at which the acceleration is zero, we consider t ≥ 0.
For cosine to be zero, t must be π/2, 3π/2, 5π/2, and so on.
Among these, the first time (t ≥ 0) when the acceleration is zero is t = π/2.
Therefore, the acceleration of the weight is zero for the first time when t = π/2.