Questions: Two masses m1=3.00 kg and m2=4.00 kg are suspended from a system of strings attached to the ceiling as shown in the figure below. Determine the magnitude of the tensions T1 and T2. Pay close attention to what the angles are measured with respect to.

Two masses m1=3.00 kg and m2=4.00 kg are suspended from a system of strings attached to the ceiling as shown in the figure below. Determine the magnitude of the tensions T1 and T2. Pay close attention to what the angles are measured with respect to.

Transcript text: Two masses $m_{1}=3.00 \mathrm{~kg}$ and $m_{2}=4.00 \mathrm{~kg}$ are suspended from a system of strings attached to the ceiling as shown in the figure below. Determine the magnitude of the tensions $T_{1}$ and $T_{2}$. Pay close attention to what the angles are measured with respect to.

Solution

Solution Steps

Step 1: Identify the forces acting on the system

The masses $ m_1 $ and $ m_2 $ are subjected to gravitational forces $ m_1 g $ and $ m_2 g $ respectively.
The tensions $ T_1 $ and $ T_2 $ are acting at angles of 30° and 40° with the horizontal.
The tension $ T_3 $ is acting vertically upward, balancing the weight of both masses.

Step 2: Calculate the total weight

The total weight $ W $ is the sum of the weights of $ m_1 $ and $ m_2 $: \[ W = m_1 g + m_2 g \] \[ W = (3.00 \, \text{kg} \times 9.8 \, \text{m/s}^2) + (4.00 \, \text{kg} \times 9.8 \, \text{m/s}^2) \] \[ W = 29.4 \, \text{N} + 39.2 \, \text{N} \] \[ W = 68.6 \, \text{N} \]

Step 3: Set up equilibrium equations

For vertical equilibrium: \[ T_1 \sin(30^\circ) + T_2 \sin(40^\circ) = W \] \[ T_1 \sin(30^\circ) + T_2 \sin(40^\circ) = 68.6 \, \text{N} \] \[ 0.5 T_1 + 0.6428 T_2 = 68.6 \]
For horizontal equilibrium: \[ T_1 \cos(30^\circ) = T_2 \cos(40^\circ) \] \[ T_1 \times 0.866 = T_2 \times 0.766 \] \[ 0.866 T_1 = 0.766 T_2 \] \[ T_1 = \frac{0.766}{0.866} T_2 \] \[ T_1 = 0.884 T_2 \]

Step 4: Solve the system of equations

Substitute $ T_1 = 0.884 T_2 $ into the vertical equilibrium equation: \[ 0.5 (0.884 T_2) + 0.6428 T_2 = 68.6 \] \[ 0.442 T_2 + 0.6428 T_2 = 68.6 \] \[ 1.0848 T_2 = 68.6 \] \[ T_2 = \frac{68.6}{1.0848} \] \[ T_2 \approx 63.23 \, \text{N} \]
Now, find $ T_1 $: \[ T_1 = 0.884 \times 63.23 \] \[ T_1 \approx 55.88 \, \text{N} \]