1)A picture of width 40 cm weighing 40N 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 3 cm higher than the points where the wire is attached to the frame. Find the tension of the wire.
the wire on each side supports mg/2
Looking at the force diagram, sinTheta(angle of depression)=(mg/2)/Tension
but tanTheta=3/20 according to the dimensions.
Tension= 20/sinTheta= 20 /sin(arctan.15)
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To find the tension of the wire, we need to analyze the forces acting on the picture.
Let's denote the tension in the wire as T.
First, let's consider the weight of the picture. The weight of the picture is given as 40N, and it acts vertically downwards.
Next, let's consider the tension in the wire. The wire is attached to the sides of the picture frame and passes over a nail. At the points where the wire is attached to the frame, the tension in the wire acts horizontally towards the center of the picture. At the nail, the tension acts upwards at an angle due to the difference in height.
To solve the problem, we can break down the forces acting on the picture frame. Let's consider the horizontal and vertical components separately.
In the horizontal direction, the tension in the wire on each side of the frame will cancel each other out because they act in opposite directions.
In the vertical direction, the sum of the upward force due to the tension in the wire and the downward force due to the weight of the picture must balance each other.
Let's solve for the tension in the wire using these equations:
Vertical Force Equilibrium:
T * cos(theta) + T * cos(theta) = 40N (where theta is the angle between the wire and the vertical line)
Since the length of the wire is not given, we can use trigonometry to determine the lengths of the horizontal and vertical sides of the right triangle formed by the wire and the difference in height of the nail.
Using Pythagorean theorem:
(40cm/2)^2 + (3cm)^2 = L^2
Simplifying the equation:
400 + 9 = L^2
L^2 = 409
L ≈ 20.22 cm
Now, we can find the value of theta:
sin(theta) = (3cm / 20.22 cm)
theta ≈ 8.885 degrees
Plugging in the value of theta into the vertical force equilibrium equation:
2T * cos(8.885 degrees) = 40N
T ≈ 40N / (2 * cos(8.885 degrees))
T ≈ 19.65N
Therefore, the tension in the wire is approximately 19.65N.