A dental bracket exerts a horizontal force of 77.7 N on a tooth at point B in the figure. What is the torque on the root of the tooth about point A?


An illustration of a tooth is shown. The tooth originates from point B and points up and to the right making an angle 48.0° with the vertical. The tip of the tooth is embedded into a piece of Gum at point A which is a distance of 1.20 cm from point B. An arrow labeled vector F acts on the bottom of the tooth at B and points to the right.


To find the torque (\(\tau\)) on the root of the tooth about point A, we need to use the definition of torque:

\[ \tau = r \times F \]

where:
- \( r \) is the lever arm (the distance from point A to point B)
- \( F \) is the force applied
- \( \theta \) is the angle between the lever arm and the force vector

In this problem:
- The force \( F \) is 77.7 N.
- The distance \( r \) from point A to point B is 1.20 cm, which we need to convert to meters (1.20 cm = 0.012 m).
- The angle \( \theta \) between the line connecting points A and B and the force vector is 48.0°.

Since the force vector is horizontal and the line from A to B makes an angle of 48.0° with the vertical, the angle between the force vector and the lever arm is:

\[ \theta = 90^\circ - 48.0^\circ = 42.0^\circ \]

The torque is given by:

\[ \tau = r \cdot F \cdot \sin(\theta) \]

Plugging in the values:

\[ \tau = 0.012 \, \text{m} \cdot 77.7 \, \text{N} \cdot \sin(42.0^\circ) \]

First, calculate \(\sin(42.0^\circ)\):

\[ \sin(42.0^\circ) \approx 0.669 \]

Now, calculate the torque:

\[ \tau = 0.012 \, \text{m} \cdot 77.7 \, \text{N} \cdot 0.669 \]
\[ \tau \approx 0.012 \cdot 77.7 \cdot 0.669 \]
\[ \tau \approx 0.621 \, \text{N} \cdot \text{m} \]

So, the torque on the root of the tooth about point A is approximately \( 0.621 \, \text{N} \cdot \text{m} \).

Would you like further details or have any questions?

### Additional Questions:
1. What would happen to the torque if the force was applied at a different angle?
2. How does the torque change if the distance between points A and B increases?
3. What if the angle between the force vector and the lever arm was 90 degrees?
4. Can we calculate the torque if the force was applied at an angle other than horizontal?
5. How would the torque change if the force applied was doubled?
6. What if the force was applied at point A instead of point B?
7. How does the direction of the force affect the torque produced?
8. Can you describe how torque affects the stability of the tooth in this scenario?

### Tip
Always ensure to convert units to the SI system (meters, Newtons, etc.) before calculating physical quantities to maintain consistency and accuracy.

Learn how to calculate torque on a tooth root when a horizontal force of 77.7 N is applied at point B, making an angle of 48.0° with the vertical. This problem involves applying the torque formula and trigonometric principles. Suitable for high school students studying physics or mechanics.

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