A weight oscillates in a vertical motion according to the position function y(t)=-5cos t. Assuming t>=0 (greater than or equal to), when will the acceleration of the weight be zero for the first time?
To find when the acceleration of the weight is zero, we need to differentiate the position function twice with respect to time.
First, let's find the velocity function by differentiating the position function once:
v(t) = dy(t)/dt
Differentiating the position function, we get:
v(t) = -5(-sin t) = 5sin t
Next, let's find the acceleration function by differentiating the velocity function:
a(t) = dv(t)/dt
Differentiating the velocity function, we get:
a(t) = 5(cos t)
Now we have the acceleration function. To find when the acceleration is zero for the first time, we need to solve the equation a(t) = 0.
Setting a(t) = 0 and solving for t, we get:
5(cos t) = 0
Since cosine is zero at t = π/2 + nπ, where n is an integer, we can find the first positive solution by setting n = 0:
t = π/2
Therefore, the acceleration of the weight will be zero for the first time at t = π/2 when t >= 0.