a body of mass 700g hangs from the end of a long wire fixed to a high tree, a horizontal string attached to the body pulls it until the wire is at 30 degree to the string .find the tension of both the string and wire
Solve the question
To find the tension in both the string and the wire, we can decompose the forces acting on the body and solve for each tension separately. Let's consider the following forces:
1. The weight of the body (mg), where m is the mass and g is the acceleration due to gravity.
2. The tension in the string.
3. The tension in the wire.
First, let's find the weight of the body:
Weight (mg) = mass (m) * acceleration due to gravity (g)
Given that the mass of the body (m) is 700 grams (or 0.7 kg) and the acceleration due to gravity (g) is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 0.7 kg * 9.8 m/s^2 = 6.86 N
Next, we need to decompose the weight force into two components. The vertical component is balanced by the tension in the wire, while the horizontal component is balanced by the tension in the string.
The horizontal component can be found using trigonometry. Since the wire makes a 30-degree angle with the string, the horizontal component of the weight force is:
Horizontal component = weight * cos(angle)
Using the angle in radians (30 degrees = (30 * pi) / 180 radians) and calculating the cosine value:
Horizontal component = 6.86 N * cos((30 * pi) / 180) = 5.937 N
Therefore, the tension in the string is equal to the horizontal component of the weight:
Tension in the string = 5.937 N
To find the tension in the wire, we look at the vertical component of the weight force, which is equal to the tension in the wire:
Tension in the wire = weight * sin(angle)
Using the same angle (in radians):
Tension in the wire = 6.86 N * sin((30 * pi) / 180) = 3.43 N
Therefore, the tension in the wire is 3.43 N.
In summary, the tension in the string is approximately 5.937 N, and the tension in the wire is approximately 3.43 N.
To find the tension in both the string and the wire, we need to analyze the forces acting on the body.
Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram of the situation. It should show the long wire, the body hanging from it, and the horizontal string attached to the body.
Step 2: Identify the forces
Identify the forces acting on the body. In this case, there are two forces: the tension in the wire (T_wire) and the tension in the string (T_string).
Step 3: Resolve the forces
Resolve the forces into their components. We can break the weight of the body (mg) into two components: one parallel to the wire (mg_parallel) and one perpendicular to the wire (mg_perpendicular).
mg_parallel = mg * sin(30°)
mg_perpendicular = mg * cos(30°)
Step 4: Equilibrium condition
Since the body is in equilibrium (not accelerating), the sum of the forces in both the vertical and horizontal directions must be zero.
In the vertical direction, the tension in the wire (T_wire) and the tension in the string (T_string) must balance the weight of the body.
T_wire + T_string = mg_parallel
In the horizontal direction, there is no net force, so the tension in the string (T_string) balances the horizontal component of the weight.
T_string = mg_perpendicular
Step 5: Substitute values and solve
Now we can substitute the values into the equations and solve them. Given:
mass of the body (m) = 700 g = 0.7 kg
angle (θ) = 30°
mg_parallel = mg * sin(30°) = 0.7 kg * 9.8 m/s^2 * sin(30°)
mg_perpendicular = mg * cos(30°) = 0.7 kg * 9.8 m/s^2 * cos(30°)
T_wire + T_string = 0.7 kg * 9.8 m/s^2 * sin(30°)
T_string = 0.7 kg * 9.8 m/s^2 * cos(30°)
Now you can calculate the tension in both the wire and the string using these equations.