ClearAll[theta, t, omega, th0, tau, expectationTheta, display, varianceTheta, p0, p, expectationPTheta, sigma, l, variancePTheta, m, g, hbar]

$Assumptions=(And@@(#>0&)/@{hbar, m, g, tau, th0, omega, sigma}) ;
display = {th0 -> (Subscript[θ, 0]),tau -> τ, omega -> ω, sigma -> σ, l -> "L", hbar -> ℏ} ;
theta[t] = th0 Cosh[ omega t ] ;
expectationTheta[n_] := (1/tau) Integrate[ (theta[t])^n, {t, 0, tau}] ;
varianceTheta = (expectationTheta[2] - expectationTheta[1]^2 // Simplify) ;
varianceTheta/. display // TraditionalForm
deltaTheta = Series[varianceTheta, {tau, 0, 4}] // Normal ;
deltaTheta/. display // TraditionalForm

p0 = (1/3) sigma l^3 th0 omega ;
p[t] = p0 Sinh[omega t] ;

expectationPTheta[n_] := (1/tau) Integrate[ (p[t])^n, {t, 0, tau}] ;
variancePTheta = (expectationPTheta[2] - expectationPTheta[1]^2 // Simplify) ;
variancePTheta/. display // TraditionalForm
deltaP = Series[variancePTheta, {tau, 0, 2}] // Normal ;
deltaP/. display // TraditionalForm

uncertainty =(deltaTheta deltaP /. {omega -> Sqrt[3g/(2l)], sigma -> m/l} // Sqrt // Simplify);
uncertainty/. display // TraditionalForm

Solve[ uncertainty == hbar/2, {tau}] /. display

(((-2))^(2/3))// N

9/5/16/108

(1/960)// Sqrt