Q: A 3.2 kg flagpole extends 1.6 m from a wall at an angle of $$25^\circ$$ above the horizontal. What is the gravitational torque about the point where the pole attaches to the wall?

$$\tau = -(1.45\,\text{m})(3.2\,\text{kg})(9.8\,\text{m/s}^2)$$

Q: Two children, Emma (25 kg) and Noah (22 kg), sit on opposite sides of a seesaw. Noah is 1.6 m from the pivot. How far from the pivot should Emma sit to balance the seesaw?

$$x = \frac{22}{25} \times 1.6\,\text{m} = 1.4\,\text{m}$$

Q: A caber (uniform rod) of length $$L$$ and mass $$M is $$pivoted at one end and released from rest at an angle $$\theta$$ from vertical. What is the initial angular acceleration $$\alpha$$?

$$\alpha = \frac{MgL \sin \theta}{I}$$, with $$I = \frac{1}{3}ML^2$

Question 1

A uniform rod of length $$L$$ and mass $$M is $$pivoted at one end and held horizontally. What is the magnitude of the gravitational torque about the pivot due to the rod's own weight?

Accepted Answer

$$\frac{1}{2}MgL \sin 90^\circ = \frac{1}{2}MgL$$

Question 2

A door is pushed with a force $$F at a $$distance $$r$$ from the hinge, making an angle $$\phi$$ with the radial line. Which equation correctly gives the magnitude of the torque?

Accepted Answer

$$	au = rF \sin \phi$$

Question 3

A 3.2 kg flagpole extends 1.6 m from a wall at an angle of $$25^\circ$$ above the horizontal. What is the gravitational torque about the point where the pole attaches to the wall?

Accepted Answer

$$	au = -(1.45\,	ext{m})(3.2\,	ext{kg})(9.8\,	ext{m/s}^2)$$

Question 4

Two children, Emma (25 kg) and Noah (22 kg), sit on opposite sides of a seesaw. Noah is 1.6 m from the pivot. How far from the pivot should Emma sit to balance the seesaw?

Accepted Answer

$$x = \frac{22}{25} 	imes 1.6\,	ext{m} = 1.4\,	ext{m}$$

Question 5

Which of the following statements about the moment of inertia is correct?

Accepted Answer

It depends on both the mass and how the mass is distributed relative to the axis of rotation.

Question 6

A high-wire walker uses a long pole to increase stability. Why does the pole help?

Accepted Answer

It increases the moment of inertia, reducing angular acceleration.

Question 7

A 5.9 m pole of mass 79 kg is pivoted at one end. What is its moment of inertia about the pivot?

Accepted Answer

$$I = \frac{1}{3}ML^2$ = \frac{1}{3}(79)(5.9^2$)$$

Question 8

A bucket of mass $$m is $$attached to a rope wound around a cylinder of mass $$M$$ and radius $$R. If $$the rope does not slip, what is the relationship between the bucket's downward acceleration $$a_y$$ and the cylinder's angular acceleration $$\alpha$$?

Accepted Answer

$$a_y = -\alpha R$$

Question 9

A wheel of radius $$R$$ rolls without slipping. What is the velocity of the center of the wheel in terms of angular velocity $$\omega$$?

Accepted Answer

$$v = \omega R$$

Question 10

A force $$F is $$applied at a distance $$r$$ from the axis, but at an angle $$30^\circ to $$the radial line. What is the moment arm for this force?

Accepted Answer

$$r \sin 30^\circ$$

Question 11

A uniform solid cylinder of mass $$M$$ and radius $$R$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2}MR^2$

Question 12

A caber (uniform rod) of length $$L$$ and mass $$M is $$pivoted at one end and released from rest at an angle $$	heta$$ from vertical. What is the initial angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{MgL \sin 	heta}{I}$$, with $$I = \frac{1}{3}ML^2$

Question 13

Which of the following best describes the center of gravity of an object?

Accepted Answer

It is the point where the entire weight of the object can be considered to act.

Question 14

A satellite of mass $$m$$ orbits a planet of mass $$M at $$radius $$r. $$What is the speed $$v of $$the satellite?

Accepted Answer

$$v = \sqrt{\frac{GM}{r}}$$

Question 15

If a net torque $$	au$$ acts on an object with moment of inertia $$I$$, what is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{	au}{I}$$

Question 16

A wheel rolls without slipping. What is the velocity of a point at the top of the wheel relative to the ground?

Accepted Answer

$$v_{top} = 2v_{center}$$

Question 17

Which of the following is true about the torque due to gravity when the pivot is at the center of gravity?

Accepted Answer

The torque is zero.

Question 18

A rope unwinds from a pulley of radius $$R$$ and moment of inertia $$I. $$The linear acceleration $$a_y of $$the object tied to the rope is related to the angular acceleration $$\alpha of $$the pulley by:

Accepted Answer

$$a_y = -\alpha R$$

Question 19

Which of the following increases the moment of inertia of a system the most?

Accepted Answer

Increasing the mass far from the axis of rotation.

Rotational Dynamics: Torque, Center of Gravity, and Rolling Motion Lecture 12

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