A picture of width 43.0 cm, weighing 52.0 N, hangs from a nail by means of flexible wire attached to the sides of the picture frame. The midpoint of the wire passes over the nail, which is 2.50 cm higher than the points where the wire is attached to the frame. Find the tension in the wire.
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To find the tension in the wire, we can consider the forces acting on the picture.
First, let's draw a free-body diagram of the picture:
```
--------T--------
/ \
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|-----W-----|
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-----------------
Nail
```
Here, T represents the tension in the wire and W represents the weight of the picture.
Now, let's consider the vertical forces on the picture. We have the weight of the picture acting downward and the tension in the wire acting upward. We can write this as an equation:
T - W = 0
Since the picture weighs 52.0 N, the weight of the picture is 52.0 N. Substituting this value into the equation:
T - 52.0 N = 0
Next, let's consider the horizontal forces on the picture. We can ignore any frictional forces since the wire is assumed to be flexible.
Since the midpoint of the wire passes over the nail, the wire will be pulled downward at the midpoint, causing a vertical component of tension in the wire. However, since the wire is attached to the sides of the picture frame, the horizontal component of tension on both sides of the picture will be equal and oppositely directed. These horizontal components will cancel each other out, resulting in a net force of 0 in the horizontal direction.
Therefore, the tension in the wire is only acting vertically.
Now, let's consider the length of the wire. Since the midpoint of the wire passes over the nail, the length of the wire can be considered as the hypotenuse of a right triangle, with the vertical distance from the nail to where the wire is attached to the frame as the opposite side, and the horizontal distance between the attachment points as the adjacent side.
Given that the nail is 2.50 cm higher than the points where the wire is attached to the frame, we can use this information to find the length of the wire.
Using the Pythagorean theorem, we have:
(length of wire)^2 = (horizontal distance)^2 + (vertical distance)^2
(length of wire)^2 = (43.0 cm)^2 + (2.50 cm)^2
(length of wire)^2 = 1850.25 cm^2 + 6.25 cm^2
(length of wire)^2 = 1856.50 cm^2
Taking the square root of both sides:
length of wire = √(1856.50 cm^2)
length of wire = 43.03 cm (approximately)
Now that we have the length of the wire, we can find the tension in the wire using the equation T - W = 0:
T - 52.0 N = 0
T = 52.0 N
Therefore, the tension in the wire is 52.0 N.