Calculate the forces F(y) associated with the following one-dimensional potential energies:
(a) $U = - ω y$
(b) $U = a y^{3} - b y^{2}$
(c) $U = U_{0} s i n β γ$

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Solution

$(a) = Given U = - ω y$

$∴ F (y) = - \frac{d U}{d y} = ω$

$(b)$
Given that
$U (y) = a y^{3} - b y^{2}$

$∴ F (y) = - \frac{d U}{d y} = 2 b y - 3 a y^{2}$

$(c)$
Given that
$U (y) = U_{0} s i n β γ$

$∴ F (y) = - \frac{d U}{d y} = 0$

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Calculate the forces F(y) associated with the following one-dimensional potential energies:(a) U=−ωy(b) U=ay3−by2(c) U=U0sinβγ

(a)=Given U=−ωy ∴F(y)=−dUdy=ω(b)Given thatU(y)=ay3−by2 ∴F(y)=−dUdy=2by−3ay2(c)Given that U(y)=U0sinβγ ∴F(y)=−dUdy=0

Calculate the forces F(y) associated with the following one-dimensional potential energies:
(a) $U = - ω y$
(b) $U = a y^{3} - b y^{2}$
(c) $U = U_{0} s i n β γ$

$(a) = Given U = - ω y$

$∴ F (y) = - \frac{d U}{d y} = ω$

$(b)$
Given that
$U (y) = a y^{3} - b y^{2}$

$∴ F (y) = - \frac{d U}{d y} = 2 b y - 3 a y^{2}$

$(c)$
Given that
$U (y) = U_{0} s i n β γ$

$∴ F (y) = - \frac{d U}{d y} = 0$