91Ó°ÊÓ

A car travels at$\mathbf{80}\text{ }\mathbf{km}\mathbf{/}\mathbf{h}\mathbf{}$on a level road in the positive direction of an xaxis. Each tire has a diameter of$66\text{ c³¾.}$Relative to a woman riding in the car, and in unit-vector notation,whatis the velocity$\stackrel{→}{v}$at the (a) center (b) top, and (c) bottom of the tire? What is the magnitude aof the acceleration at the (d) center (e)top, and (f)the bottom of each tire? Relative to a hitchhiker sitting next to the road and in unit-vector notation, what is the velocity$\stackrel{→}{v}$at the (g) centre (h) top, and (i) bottom of the tire? And the magnitude aof the acceleration at the (j) centre (k) top(l) bottom of each tire?

Figure 11-23 shows three particles of the same mass and the same constant speed moving as indicated by the velocity vectors. Points a, b, c, and dform a square, with point eat the center. Rank the points according to the magnitude of the net angular momentum of the three-particle system when measured about the points, greatest first.

A plum is located at coordinates (-2.0M, 0,4.0M). In unit- vector notation, what is the torque about the origin on the plum if that torque is due to a force $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}$. (a) Whose only component is role="math" localid="1661237442203" ${\mathit{F}}_{\mathbf{x}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathit{N}$? (b)Whose only component is ${\mathit{F}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{0}\mathit{N}$? (c)Whose only component is ${\mathit{F}}_z\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathit{N}$?(d) Whose only component is ${\mathit{F}}_z\mathbf{=}\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{0}\mathit{N}$?

In unit-vector notation, what is the torque about the origin on a particle located at coordinates (0, -4.0m, 3.0m) if that torque is due to

(a) Force ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{1}}$ with components F_1x = 2.0 N, F_1y = F_1z = 0,

(b) Force ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{2}}$with components F_2x = 0,F_2x = 2.0N,F_2x = 4.0N?

Question: A particle moves through an xyz coordinate system while a force acts on the particle. When the particle has the position vector $\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\text{m}}\mathbf{)}\widehat{\mathbf{\text{i}}}\mathbf{-}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\text{m}}\mathbf{)}\widehat{\mathbf{\text{j}}}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\text{m}}\mathbf{)}\widehat{\mathbf{\text{k}}}$,the force is given by $\stackrel{\mathbf{→}}{\mathbf{F}}\mathbf{=}{\mathbf{F}}_{\mathbf{x}}\widehat{\mathbf{\text{i}}}\mathbf{+}\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{00}\mathbf{\text{N}}\mathbf{)}\widehat{\mathbf{\text{j}}}\mathbf{-}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{\text{N}}\mathbf{)}\widehat{\mathbf{\text{k}}}$and the corresponding torque about the origin is$\stackrel{\mathbf{→}}{\mathbf{τ}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{\text{Nm}}\mathbf{)}\widehat{\mathbf{\text{i}}}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\text{Nm}}\mathbf{)}\widehat{\mathbf{\text{j}}}\mathbf{-}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{\text{Nm}}\mathbf{)}\widehat{\mathbf{\text{k}}}$. Determine.

Force $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}\mathbf{=}\left(2.0N\right)\hat{\mathbf{i}}\mathbf{-}\left(3.0N\right)\hat{\mathbf{k}}$acts on a pebble with position vector$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{r}}\mathbf{=}\left(0.50m\right)\hat{\mathbf{j}}\mathbf{-}\left(2.0m\right)\hat{\mathbf{k}}$relative to the origin. In unit- vector notation, (a) What is the resulting torque on the pebble about the origin and (b) What is the resulting torque on the pebble about the point$\left(2.0m,0,-3.0m\right)\mathbf{?}$

In unit-vector notation, what is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0m,-2.0m,4.0m) due to (a) force ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{1}}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{i}}\mathbf{-}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{k}}$,(b) force ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{2}}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{i}}\mathbf{-}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{k}}$, (c) the vector sum of ${\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{1}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}{\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{2}}$? (d) Repeat part (c) for the torque about the point with coordinates (3.0m, 2.0m, 4.0m).

Force$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{N}\mathbf{)}\hat{\mathbf{j}}$acts on a particle with position vector$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{r}}\mathbf{=}\left(3.0m\right)\hat{\mathbf{i}}\mathbf{+}\left(4.0m\right)\hat{\mathbf{j}}$. (a) What is the torque on the particle about the origin, in unit-vector notation? (b) What is the angle between the directions of$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{r}}$ and$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}$?

At the instant of Figure, a 2.0kg particle Phas a position vector $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{r}}$ of magnitude 3.0mand angle ${\mathit{θ}}_{\mathbf{1}}\mathbf{=}{\mathbf{45}}^{\mathbf{o}}$ and a velocity vector$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$ of magnitude 4.0m/sand angle${\mathit{θ}}_{\mathbf{2}}\mathbf{=}{\mathbf{30}}^{\mathbf{0}}$. Force$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}$, of magnitude 2.0Nand angle${\mathit{θ}}_{\mathbf{3}}\mathbf{=}{\mathbf{30}}^{\mathbf{o}}$acts on P. All three vectors lie in the xy plane. About the origin, (a) What is the magnitude of the angular momentum of P? (b) What is the direction of the angular momentum of P? (c) What is the magnitude of the torque acting on P? (d)What is the direction of the torque acting on P?

A 2.0kg particle-like object moves in a plane with velocity components v_x = 30m/sand v_y = 60m/sas it passes through the point with (x,y)coordinates of (3.0, -4.0)m.Just then, in unit-vector notation, (a) what is its angular momentum relative to the origin and (b) what is its angular momentum relative to the point located at (-2.0, -2.0)m?