A 10-kg mass is suspended by two strings of length 5m and 7m attached to two points in the ceiling 10 m apart. Determine the tension on each string. Round your result to one decimal place (include a vector and position diagrams)
To determine the tension on each string, we need to analyze the forces acting on the mass. Let's break down the problem step by step:
1. Draw a diagram: Draw a diagram representing the situation. You can label the mass as M and the two points in the ceiling as A and B. Label the length of the first string as L1 (5m) and the length of the second string as L2 (7m).
A-----5m-----M------7m------B
2. Analyze the forces: Identify the forces acting on the mass. In this case, there are two tension forces from the strings pulling upwards, and the gravitational force acting downwards.
3. Resolve the forces: Resolve the forces into horizontal and vertical components. Since the mass is in equilibrium, the vertical components of the tension forces must balance the gravitational force, and the horizontal components of the tension forces must cancel each other out.
4. Write the equations: Let's assume that the angles made by the strings with the vertical axis are θ1 and θ2 (refer to the diagram). The forces can be resolved as:
Vertical components:
T1 * cos(θ1) + T2 * cos(θ2) = Mg -- (Equation 1)
Horizontal components:
T1 * sin(θ1) = T2 * sin(θ2) -- (Equation 2)
Where T1 and T2 are the tensions in the respective strings, and Mg is the gravitational force (mass * acceleration due to gravity).
5. Solve the equations: Since we have two unknowns (T1 and T2), we need to solve the equations simultaneously. Rearrange Equation 2 to solve for T1:
T1 = T2 * sin(θ2) / sin(θ1) -- (Equation 3)
Substitute Equation 3 into Equation 1:
T2 * sin(θ2) / sin(θ1) * cos(θ1) + T2 * cos(θ2) = Mg
Simplify the equation by dividing both sides by T2:
sin(θ2) * cos(θ1) + cos(θ2) = Mg / T2
Now, we can solve for T2:
T2 = Mg / (sin(θ2) * cos(θ1) + cos(θ2)) -- (Equation 4)
6. Calculate the values: Substitute the given values into Equation 4 and calculate T2.
- Mass (M) = 10 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
To find θ1 and θ2, we can use the trigonometric relationships:
- cos(θ1) = L1 / (distance between A and B)
- cos(θ2) = L2 / (distance between A and B)
Substitute the values into Equation 4, and you will have the tension T2. Then, substitute T2 back into Equation 3 to find T1.
7. Round the result: Finally, round the tensions T1 and T2 to one decimal place as requested in the question.