In an iron ore handling port, a barge is pulled by ropes using two tugboats as shown in the figure. In equilibrium, the resultant of the forces $T_1$ and $T_2$ along the axis of the barge in the direction of its travel is 5000 N. The tensions $T_1$ and $T_2$ in N respectively are
To solve this problem, we need to consider the forces acting on the barge due to the tensions \(T_1\) and \(T_2\). The resultant force along the direction of travel of the barge is given as 5000 N.
Here’s the approach to solve the problem step-by-step:
- Let \(\theta_1 = 30^\circ\) be the angle \(T_1\) makes with the direction of travel, and \(\theta_2 = 45^\circ\) be the angle \(T_2\) makes with the direction of travel.
- The horizontal components of \(T_1\) and \(T_2\) can be expressed using trigonometry:
- \(T_{1x} = T_1 \cdot \cos(30^\circ)\)
- \(T_{2x} = T_2 \cdot \cos(45^\circ)\)
- According to the problem, the sum of these horizontal components results in a force of 5000 N:
- \(T_1 \cdot \cos(30^\circ) + T_2 \cdot \cos(45^\circ) = 5000\)
- Using the cosine values, the equation becomes:
- \(T_1 \cdot \frac{\sqrt{3}}{2} + T_2 \cdot \frac{\sqrt{2}}{2} = 5000\)
- Along the vertical direction, the barge is in equilibrium since there is no vertical movement, thus:
- \(T_1 \cdot \sin(30^\circ) = T_2 \cdot \sin(45^\circ)\)
- Using the sine values, we have:
- \(T_1 \cdot \frac{1}{2} = T_2 \cdot \frac{\sqrt{2}}{2}\)
- Solving these equations simultaneously:
- From \(T_1 \cdot \frac{\sqrt{3}}{2} + T_2 \cdot \frac{\sqrt{2}}{2} = 5000\), and
- \(T_1 \cdot \frac{1}{2} = T_2 \cdot \frac{\sqrt{2}}{2}\), leading to \(T_1 = T_2 \sqrt{2}\).
- Substituting \(T_1 = T_2 \sqrt{2}\) in the first equation:
- \((T_2 \sqrt{2}) \cdot \frac{\sqrt{3}}{2} + T_2 \cdot \frac{\sqrt{2}}{2} = 5000\)
- After simplifying, we get \(T_2 = 2588\) N and \(T_1 = 3660\) N.
Thus, the tensions are 3660 N and 2588 N respectively.
The correct answer is: 3660 and 2588.
Important Questions from Vector Algebra
What is the length of projection of the vector \(\rm \hat{i}+2 \hat{j}+3 \hat{k}\) on the vector \(\rm2 \hat{i}+3 \hat{j}-2 \hat{k}\) ?
Consider the following in respect of the vectors \(\rm \vec{a}=(0,1,1)\) and \(\rm \vec{b}=(1,0,1) \) :
1. The number of unit vectors perpendicular to both \(\rm \vec{a}\) and \(\rm \vec{b}\) is only one.
2. The angle between the vectors is \(\frac{\pi}{3}\).
Which of the statements given above is/are correct?
Consider the following points :
1. (-1, -3, 1)
2. (-1, 3, 2)
3. (-2, 5, 3)
Which of the above points lie on the line joining A and B ?
What is the magnitude of \(\overrightarrow{A B}\) ?
If \({\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) , then which of the following equations is/are correct?
1. y – x = 4
2. 2z – 3 = 0
Select the correct answer using the code given below:
