Q: A particle is located at the vector position $$\vec{r} = (i + 5j)\ \mathrm{m}$$, and the force acting on it is $$\vec{F} = (3i + 3j)\ \mathrm{N}. $$What is the torque about the origin?

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Q: A conical pendulum consists of a bob of mass $$m in $$motion in a horizontal circle of radius $$r at $$constant speed. The supporting wire of length $$\ell$$ makes a constant angle $$\theta$$ with the vertical. What is the correct expression for the magnitude of the angular momentum about the center of the circle?

$$L = \left( m g^2$ \$$ell^3$$ \sin^4$ \theta / \cos \theta \$$right)^{1/2}$$

Q: A particle is located at $$\vec{r} = (i + 5j)\ \mathrm{m}$$ and a force $$\vec{F} = (3i + 3j)\ \mathrm{N}$$ acts on it. What is the $$z-$$component of the torque about the origin?

$$\tau_z = r_x F_y - r_y F_x = 1 \times 3 - 5 \times 3 = -12\ \mathrm{N \cdot m}$$

Question 1

A particle of mass $$m$$ moves in a circular path of radius $$r in $$the $$xy-$$plane with constant speed $$v. $$What is the magnitude of its angular momentum about the center of the circle?

Accepted Answer

$$L = mvr$$

Question 2

A bowling ball of mass $$m = 100\ \mathrm{g}$$ and radius $$R = 10\ \mathrm{cm}$$ spins at $$10\ \mathrm{rev/s}. $$What is the magnitude of its angular momentum about its center? (Assume the ball is a solid sphere.)

Accepted Answer

$$L = \frac{2}{5}mR^2$ \omega$$

Question 3

A light, rigid rod $$1.00\ \mathrm{m} in $$length joins two particles with masses $$4.00\ \mathrm{kg}$$ and $$3.00\ \mathrm{kg} at $$its ends. The combination rotates in the $$xy-$$plane about a pivot through the center of the rod. If the speed of each particle is $$5.00\ \mathrm{m/s}$$, what is the total angular momentum of the system about the origin?

Accepted Answer

$$L = m_1$ v r_1$ + m_2$ v r_2$

Question 4

A particle is located at the vector position $$\vec{r} = (i + 5j)\ \mathrm{m}$$, and the force acting on it is $$\vec{F} = (3i + 3j)\ \mathrm{N}. $$What is the torque about the origin?

Accepted Answer

$$\vec{	au} = \vec{r} 	imes \vec{F}$$

Question 5

A conical pendulum consists of a bob of mass $$m in $$motion in a horizontal circle of radius $$r at $$constant speed. The supporting wire of length $$\ell$$ makes a constant angle $$	heta$$ with the vertical. What is the correct expression for the magnitude of the angular momentum about the center of the circle?

Accepted Answer

$$L = \left( m g^2$ \$$ell^3$$ \sin^4$ 	heta / \cos 	heta \$$right)^{1/2}$$

Question 6

A particle moves in a circle of radius $$r$$ with angular speed $$\omega. $$What is the relationship between the magnitude of its angular momentum $$L$$ and $$\omega$$?

Accepted Answer

$$L = m r^2$ \omega$$

Question 7

A force $$\vec{F}$$ acts on a particle at position $$\vec{r}. $$Which of the following best describes the physical meaning of the torque $$\vec{	au} = \vec{r} 	imes \vec{F}$$?

Accepted Answer

It measures the tendency of the force to cause rotation about the origin.

Question 8

A rod of length $$L$$ rotates about its center with two masses $$m_1$$ and $$m_2 at $$its ends. If $$m_1$$ \u003e $$m_2$$, which statement is true about their contributions to the total angular momentum?

Accepted Answer

The heavier mass contributes more to the total angular momentum.

Question 9

A particle is located at $$\vec{r} = (i + 5j)\ \mathrm{m}$$ and a force $$\vec{F} = (3i + 3j)\ \mathrm{N}$$ acts on it. What is the $$z-$$component of the torque about the origin?

Accepted Answer

$$	au_z = r_x F_y - r_y F_x = 1 	imes 3 - 5 	imes 3 = -12\ \mathrm{N \cdot m}$$

Question 10

A conical pendulum has a bob of mass $$m$$ attached to a string of length $$\ell$$ making an angle $$	heta$$ with the vertical. If the tension in the string is $$T$$, what is the vertical component of the tension?

Accepted Answer

$$T \cos 	heta$$

Question 11

A bowling ball (solid sphere) of mass $$m$$ and radius $$R$$ spins with angular speed $$\omega. $$Which formula gives its moment of inertia about its center?

Accepted Answer

$$I = \frac{2}{5} m R^2$

Question 12

A particle moves in a circle of radius $$r$$ with constant speed $$v. $$What is the direction of its angular momentum vector $$\vec{L}$$ about the center?

Accepted Answer

Perpendicular to the plane of motion (along the axis of rotation)

Question 13

A system consists of two masses, $$m_1$$ and $$m_2$$, attached to the ends of a light rod of length $$L$$ rotating about its center. If $$m_1 = 4$$\ \mathrm{kg}$$, m_2$ = 3\ \mathrm{kg}$$, and $$L = 1\ \mathrm{m}$$, what is the total moment of inertia about the center?

Accepted Answer

$$I = m_1$ (\frac{L}{2})^2$ + m_2$ (\frac{L}{2})^2$

Question 14

A conical pendulum bob of mass $$m$$ moves in a horizontal circle of radius $$r at $$constant speed $$v. If $$the string makes an angle $$	heta$$ with the vertical, what is the expression for $$r in $$terms of $$\ell$$ and $$	heta$$?

Accepted Answer

$$r = \ell \sin 	heta$$

Question 15

A force $$\vec{F}$$ acts at a point with position vector $$\vec{r}$$ relative to the origin. If $$\vec{F} is $$parallel to $$\vec{r}$$, what is the torque about the origin?

Accepted Answer

Zero

Question 16

A particle of mass $$m$$ moves in a circle of radius $$r$$ with speed $$v. If $$the speed is doubled, how does the angular momentum change?

Accepted Answer

It doubles

Question 17

A rod of length $$L$$ with masses $$m_1$$ and $$m_2 at $$its ends rotates about its center. If the angular speed is $$\omega$$, what is the total angular momentum?

Accepted Answer

$$L = [m_1$ (\frac{L}{2})^2$ + m_2$ (\frac{L}{2})^2$] \omega$$

Angular Momentum and Rotational Motion: Study Notes

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