Question

10.1.1 An oversimplified dumbbell Below is a dumbbell supported by a pivot (i.e., the dot in the middle of the rod) on a horizontal tabletop. The dumbbell can be treated as three parts putting together: two heavy balls with radius $R$ and mass $M_b$, and one rod with length $L$ and mass $M_r$. Now three forces are exerted on the dumbbell. Assuming the positive $z$ direction is perpendicular to the page and pointing outward, please derive a symbolic expression for the net torque.\\ $\vec{F}_a = \vec{F}$ \quad $\vec{F}_b = \vec{F}$ \\ $\vec{\tau}_a = \vec{r}_a \times \vec{F}_a = (-\frac{L}{2}\hat{i}) \times (+F\hat{j}) = (-\frac{FL}{2})\hat{k}$ \\ $\vec{\tau}_b = \vec{r}_b \times \vec{F}_b = (+\frac{L}{4}\hat{i}) \times ((-F\cos 45^\circ)\hat{i} + (+F\sin 45^\circ)\hat{j})$ \\ $= (+\frac{FL}{4}\sin 45^\circ)\hat{k}$ \\ $\vec{\tau}_c = \vec{r}_c \times \vec{F}_c = 0$ \\ $\vec{\tau}_{net} = \vec{\tau}_a + \vec{\tau}_b + \vec{\tau}_c = (-\frac{FL}{2} + \frac{FL}{4}\sin 45^\circ)\hat{k}$

          10.1.1 An oversimplified dumbbell Below is a dumbbell supported by a pivot (i.e., the dot in the middle of the rod) on a horizontal tabletop. The dumbbell can be treated as three parts putting together: two heavy balls with radius $R$ and mass $M_b$, and one rod with length $L$ and mass $M_r$. Now three forces are exerted on the dumbbell. Assuming the positive $z$ direction is perpendicular to the page and pointing outward, please derive a symbolic expression for the net torque.\\
$\vec{F}_a = \vec{F}$ \quad $\vec{F}_b = \vec{F}$ \\
$\vec{\tau}_a = \vec{r}_a \times \vec{F}_a = (-\frac{L}{2}\hat{i}) \times (+F\hat{j}) = (-\frac{FL}{2})\hat{k}$ \\
$\vec{\tau}_b = \vec{r}_b \times \vec{F}_b = (+\frac{L}{4}\hat{i}) \times ((-F\cos 45^\circ)\hat{i} + (+F\sin 45^\circ)\hat{j})$ \\
$= (+\frac{FL}{4}\sin 45^\circ)\hat{k}$ \\
$\vec{\tau}_c = \vec{r}_c \times \vec{F}_c = 0$ \\
$\vec{\tau}_{net} = \vec{\tau}_a + \vec{\tau}_b + \vec{\tau}_c = (-\frac{FL}{2} + \frac{FL}{4}\sin 45^\circ)\hat{k}$

10.1.1 An oversimplified dumbbell Below is a dumbbell supported by a pivot (i.e., the dot in the middle of the rod) on a horizontal tabletop. The dumbbell can be treated as three parts putting together: two heavy balls with radius R and mass Mb, and one rod with length L and mass Mr. Now three forces are exerted on the dumbbell. Assuming the positive z direction is perpendicular to the page and pointing outward, please derive a symbolic expression for the net torque.

F⃗a = F⃗ F⃗b = F⃗

τ⃗a = r⃗a ×F⃗a = (-(L)/(2)î) × (+Fĵ) = (-(FL)/(2))k̂

τ⃗b = r⃗b ×F⃗b = (+(L)/(4)î) × ((-Fcos 45^∘)î + (+Fsin 45^∘)ĵ)

= (+(FL)/(4)sin 45^∘)k̂

τ⃗c = r⃗c ×F⃗c = 0

τ⃗net = τ⃗a + τ⃗b + τ⃗c = (-(FL)/(2) + (FL)/(4)sin 45^∘)k̂

Added by Monica T.

Question

Please give Ace some feedback