Kinetic energy in cylindrical coordinates

Classical Mechanics Level 2

It is well known that the kinetic energy of a particle in cartesian coordinates is given by:

E k i n = m 2 ∣ v → ∣ 2 \boxed{E_{kin}=\frac{m}{2}|\overrightarrow{v}|^2} where ∣ v → ∣ 2 = x ˙ 2 + y ˙ 2 + z ˙ 2 |\overrightarrow{v}|^2 = \dot{x}^2+\dot{y}^2+\dot{z}^2 .

Remember that for cylindrical coordinates we have:

x → = [ x ( t ) y ( t ) z ( t ) ] ↦ [ r ( t ) cos ( ϕ ( t ) ) r ( t ) sin ( ϕ ( t ) ) z ( t ) ] \overrightarrow{x} = \begin{bmatrix} x(t) \\[0.3em] y(t) \\[0.3em] z(t) \end{bmatrix} \mapsto \begin{bmatrix} r(t) \text{cos}(\phi(t)) \\[0.3em] r(t) \text{sin}(\phi(t))\\[0.3em] z(t) \end{bmatrix}

From this information, find the expression for E k i n E_{kin} in cylindrical coordinates.

Possible answers:

A: E k i n = m 2 ( r ˙ 2 + r 2 ϕ 2 + z ˙ 2 ) E_{kin}=\frac{m}{2}(\dot{r}^2+r^2\phi^2+\dot{z}^2)

B: E k i n = m 2 ( r ˙ 2 + r 2 ϕ ˙ 2 + z ˙ 2 ) E_{kin}=\frac{m}{2}(\dot{r}^2+r^2\dot{\phi}^2+\dot{z}^2)

C: E k i n = m 2 ( r 2 + r ˙ 2 ϕ ˙ 2 + z ˙ 2 ) E_{kin}=\frac{m}{2}(r^2+\dot{r}^2\dot{\phi}^2+\dot{z}^2)

D: E k i n = m 2 ( r ˙ 2 + r ˙ 2 ϕ ˙ 2 + z ˙ 2 ) E_{kin}=\frac{m}{2}(\dot{r}^2+\dot{r}^2\dot{\phi}^2+\dot{z}^2)

B C A D

Claudio Brot by Claudio Brot , 27, Switzerland

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1 solution

Tom Engelsman
Apr 24, 2020

Taking the time derivatives of the cylindrical coordinate vector above yields:

x ˙ ( t ) = r ˙ cos ⁡ ( ϕ ) − r ϕ ˙ sin ⁡ ( ϕ ) \dot{x}(t) = \dot{r} \cos(\phi) -r \dot{\phi} \sin(\phi) ,

y ˙ ( t ) = r ˙ sin ⁡ ( ϕ ) + r ϕ ˙ cos ⁡ ( ϕ ) \dot{y}(t) = \dot{r} \sin(\phi) +r \dot{\phi} \cos(\phi) ,

z ˙ ( t ) = z ˙ \dot{z}(t) = \dot{z} .

The kinetic energy in cylindrical coordinates is finally calculated according to:

E k i n = m 2 ∣ v ⃗ ∣ 2 = m 2 ⋅ [ ( r ˙ cos ⁡ ( ϕ ) − r ϕ ˙ sin ⁡ ( ϕ ) ) 2 + ( r ˙ sin ⁡ ( ϕ ) + r ϕ ˙ cos ⁡ ( ϕ ) ) 2 + z ˙ 2 ] ; E_{kin} = \frac{m}{2} |\vec{v}|^2 = \frac{m}{2} \cdot [( \dot{r} \cos(\phi) -r \dot{\phi} \sin(\phi))^2 + (\dot{r} \sin(\phi) +r \dot{\phi} \cos(\phi))^2 + \dot{z}^{2}];

or m 2 ⋅ [ [ r ˙ 2 cos ⁡ 2 ( ϕ ) − 2 r r ˙ ϕ ˙ cos ⁡ ( ϕ ) sin ⁡ ( ϕ ) ) + r 2 ϕ ˙ 2 sin ⁡ 2 ( ϕ ) ] + [ r ˙ 2 sin ⁡ 2 ( ϕ ) + 2 r r ˙ ϕ ˙ cos ⁡ ( ϕ ) sin ⁡ ( ϕ ) ) + r 2 ϕ ˙ 2 cos ⁡ 2 ( ϕ ) ] + z ˙ 2 ] ; \frac{m}{2} \cdot [ [\dot{r}^{2} \cos^{2}(\phi) -2r\dot{r} \dot{\phi} \cos(\phi)\sin(\phi)) + r^2 \dot{\phi}^{2} \sin^{2}(\phi)] + [\dot{r}^{2} \sin^{2}(\phi) + 2r\dot{r} \dot{\phi} \cos(\phi)\sin(\phi)) + r^2 \dot{\phi}^{2} \cos^{2}(\phi)] + \dot{z}^{2}];

or m 2 ⋅ [ [ r ˙ 2 cos ⁡ 2 ( ϕ ) + r 2 ϕ ˙ 2 sin ⁡ 2 ( ϕ ) ] + [ r ˙ 2 sin ⁡ 2 ( ϕ ) + r 2 ϕ ˙ 2 cos ⁡ 2 ( ϕ ) ] + z ˙ 2 ] ; \frac{m}{2} \cdot [ [\dot{r}^{2} \cos^{2}(\phi) + r^2 \dot{\phi}^{2} \sin^{2}(\phi)] + [\dot{r}^{2} \sin^{2}(\phi) + r^2 \dot{\phi}^{2} \cos^{2}(\phi)] + \dot{z}^{2}];

or m 2 ⋅ [ r ˙ 2 [ cos ⁡ 2 ( ϕ ) + sin ⁡ 2 ( ϕ ) ] + r 2 ϕ ˙ 2 [ sin ⁡ 2 ( ϕ ) + cos ⁡ 2 ( ϕ ) ] + z ˙ 2 ] ; \frac{m}{2} \cdot [ \dot{r}^{2} [\cos^{2}(\phi) + \sin^{2}(\phi)] + r^2 \dot{\phi}^{2} [\sin^{2}(\phi) + \cos^{2}(\phi)] + \dot{z}^{2}];

or m 2 ⋅ [ r ˙ 2 + r 2 ϕ ˙ 2 + z ˙ 2 ] . \boxed{\frac{m}{2} \cdot [ \dot{r}^{2} + r^2 \dot{\phi}^{2} + \dot{z}^{2}]}.

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