At a stable equilibrium, what is the net force?

For small oscillations about equilibrium, should \(k_{\mathrm{eff}}\) be positive or negative?

A force near equilibrium is \(F(x)\approx -12\eta\), where \(\eta=x-x_0\). Find \(k_{\mathrm{eff}}\).

At an equilibrium point, \(F'(x_0)=-20\,\mathrm{N\,m^{-1}}\). Find \(k_{\mathrm{eff}}\).

For \(F(x)=6-3x\) in SI units, find the equilibrium position and \(k_{\mathrm{eff}}\).

A mass \(0.50\,\mathrm{kg}\) oscillates near equilibrium with \(k_{\mathrm{eff}}=18\,\mathrm{N\,m^{-1}}\). Find \(\omega\) and \(T\).

Near equilibrium, a force is \(F=-4(x-2)\) in SI units. For mass \(1.0\,\mathrm{kg}\), find \(x_0\) and \(\omega\).

A vertical spring has \(k=120\,\mathrm{N\,m^{-1}}\) and supports \(m=3.0\,\mathrm{kg}\). Find the static equilibrium stretch.

A vertical spring stretches \(0.080\,\mathrm{m}\) under a hanging mass. Find the small-oscillation angular frequency using \(g=9.8\,\mathrm{m\,s^{-2}}\).

A potential near equilibrium is \(U(x)=U_0+\frac{1}{2}\alpha(x-x_0)^2\). Find the small-oscillation angular frequency for mass \(m\).

A particle of mass \(m\) moves near \(x=0\) under \(F(x)=-\alpha x+\beta x^2\), with \(\alpha>0\). Find the small-oscillation angular frequency about \(x=0\).

A force is \(F(x)=ax-bx^3\), with \(a,b>0\). Determine whether \(x=0\) is a stable equilibrium for small oscillations.

A mass on a spring has net force \(F=F_0-kx\). Find the shifted equilibrium and the small-oscillation angular frequency.

A potential is \(U(x)=U_0-\frac{1}{2}cx^2\), with \(c>0\). Is \(x=0\) a stable small-oscillation equilibrium? Explain using the force slope or potential curvature.

For \(F(x)=C/x^2-D\), with \(C,D>0\), find the equilibrium position and the small-oscillation angular frequency for mass \(m\).

A force near equilibrium is \(F(\eta)=-a\eta-c\eta^3\), with \(a,c>0\). Find the leading small-amplitude angular frequency and state what part of the force changes only larger-amplitude behavior.

A particle of mass \(m\) has potential energy \(U(x)=U_0+\alpha x^2+\beta x^3+\gamma x^4\), with \(\alpha>0\). Derive the small-oscillation angular frequency about \(x=0\), and explain why the cubic and quartic terms do not affect the leading small-amplitude frequency.

A particle of mass \(m\) has force \(F(x)=p-qx+sx^2\), with \(p,q,s>0\). Derive the equilibrium positions when they exist, and determine which equilibrium is stable.

A mass \(m\) is attached to two horizontal springs fixed to walls. The left spring has constant \(k_1\) and natural length arranged so it is unstretched at \(x=0\); the right spring has constant \(k_2\) and is also unstretched at \(x=0\). Derive the small-oscillation frequency if the mass is displaced by \(x\), and then state how a constant external force \(F_0\) changes the equilibrium and the frequency.

A particle of mass \(m\) has potential \(U(x)=\alpha x^4-\beta x^2\), with \(\alpha,\beta>0\). Find all equilibrium points, classify their stability, and derive the small-oscillation angular frequency about each stable equilibrium.