a picture frame 25 kg hung over a nail by a string attatched to two hookes in the upper bar . find tension using parallelogram law
To find the tension in the string using the parallelogram law, we need to understand the concept of vector addition.
The parallelogram law states that if two vectors are represented as two adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the resultant of the two vectors.
In this scenario, the weight of the picture frame (25 kg) is acting vertically downward. Let's represent this weight as vector W.
To find the tension, we need to add two vectors: the tension in the string and the weight vector.
Now, let's break the weight vector into two components: one parallel to the string (tension) and the other perpendicular to it. The component parallel to the string is the weight of the picture frame, which is 25 kg, and the component perpendicular to the string is zero since the string does not resist any horizontal force.
Now, we have two vectors: the tension vector (T) and the weight vector (W). The angle between the two vectors is 180 degrees since they are acting in opposite directions.
Using the parallelogram law, we can find the resultant vector (R), which represents the tension in the string. The magnitude of the resultant vector R is equal to the magnitude of the vector sum of T and W, and the direction of R is opposite to that of W.
To find the tension, we can use the following formula:
R^2 = T^2 + W^2 + (2 × T × W × cos θ)
Where:
R is the magnitude of the resultant vector R
T is the magnitude of the tension vector T
W is the magnitude of the weight vector W
θ is the angle between T and W
Given that the weight of the picture frame is 25 kg, we have:
W = 25 kg
Since the angle between T and W is 180 degrees, we have:
θ = 180 degrees
Now, we can substitute the given values into the formula and solve for T:
R^2 = T^2 + (25 kg)^2 + (2 × T × 25 kg × cos 180°)
Simplifying the equation, we have:
R^2 = T^2 + 625 kg^2 + (-50T)
Since the tension T cannot be negative, we can ignore the negative term (-50T).
R^2 = T^2 + 625 kg^2
We know that the magnitude of the resultant vector R is equal to the weight of the picture frame, which is 25 kg.
Therefore, we have:
25^2 = T^2 + 625 kg^2
Solving this equation, we get:
625 = T^2 + 625 kg^2 - 625 kg^2
625 = T^2
Taking the square root of both sides, we find:
T = √625
T = 25 kg
Therefore, the tension in the string is 25 kg.