A picture hangs on the wall suspended by two strings, as shown in the figure, with θ = 59°. The tension in string 1 is 1.9 N.
To find the tension in string 2, we can use trigonometry. Let's break down the problem step by step.
Step 1: Identify the forces acting on the picture.
In this scenario, there are two forces acting on the picture: the tension in string 1, denoted as T1 and the tension in string 2, denoted as T2.
Step 2: Understand the properties of a hanging picture.
When a picture hangs on the wall, it is in equilibrium. This means that the sum of all vertical forces and the sum of all horizontal forces acting on the picture equals zero.
Step 3: Resolve the forces.
We will resolve the forces acting on the picture into vertical and horizontal components.
For string 1 (T1),
- The vertical component of T1 is T1 * sin(θ) since θ is the angle the string makes with the vertical direction.
- The horizontal component of T1 is T1 * cos(θ) since θ is the angle the string makes with the horizontal direction.
For string 2 (T2),
- The vertical component of T2 is T2 * sin(θ) since θ is the angle the string makes with the vertical direction.
- The horizontal component of T2 is T2 * cos(θ) since θ is the angle the string makes with the horizontal direction.
Step 4: Write down the equations for equilibrium.
Since the picture is in equilibrium, the sum of vertical forces and horizontal forces equals zero.
For the vertical forces,
T1 * sin(θ) + T2 * sin(θ) = 0
For the horizontal forces,
T1 * cos(θ) - T2 * cos(θ) = 0
Step 5: Solve the equations for T2.
We can rearrange the equations to solve for T2.
From the equation for vertical forces,
T2 * sin(θ) = -T1 * sin(θ)
Dividing both sides by sin(θ) gives,
T2 = -(T1 * sin(θ)) / sin(θ)
T2 = -T1
From the equation for horizontal forces,
T1 * cos(θ) - T2 * cos(θ) = 0
Substituting T2 = -T1,
T1 * cos(θ) - (-T1 * cos(θ)) = 0
T1 * cos(59°) + T1 * cos(59°) = 0
2 * T1 * cos(59°) = 0
T1 * cos(59°) = 0
Since the tension in string 1, T1, is 1.9 N and cos(59°) doesn't equal zero, we can conclude that T2 is zero.
Therefore, the tension in string 2 is zero N.