A lamp of weight 20 N hangs from two strings
Scalars & Vectors - OCR A-Level Physics
- A lamp of weight 20 N hangs from two strings. String A makes 30 degrees to the horizontal. String B makes 50 degrees to the horizontal.
- The lamp is in equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹... Find the tension in each string.
- Step 1: Draw a free body diagramA diagram showing all the forces acting on a single object, drawn as arrows from the centre of the object.. Three forces act on the lamp: weight W = 20 N downward, tension T_A along string A, tension T_B along string B.
- Step 2: Resolve horizontally. For equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.., horizontal components cancel:
- $T_A cos(30) = T_B cos(50).$
- Step 3: Resolve vertically. For equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.., vertical components support the weight:
- $T_A sin(30) + T_B sin(50) = 20.$
$$T_A \cos(30^\circ) = T_B \cos(50^\circ)$$
$$\begin{aligned}
T_A &= T_B \times \frac{\cos(50^\circ)}{\cos(30^\circ)} \\
&= T_B \times \frac{0.643}{0.866} \\
&= 0.742 \, T_B
\end{aligned}$$
- Substitute into the vertical equation:
- 0.742 T_B x sin(30) + T_B x sin(50) = 20.
- 0.742 T_B x 0.500 + T_B x 0.766 = 20.
- 0.371 T_B + 0.766 $T_B = 20$.
- 1.137 $T_B = 20$.
- $T_B = 17.6 N.$
- $T_A = 0.742 \times 17.6 = 13.1 N.$
$$\begin{aligned}
T_B &= 17.6 \text{ N}, \quad T_A \\
&= 13.1 \text{ N}
\end{aligned}$$
Examiner Tips and Tricks
- Check your answer: the string closer to vertical (B, at 50 degrees) supports more weight, which makes physical sense.
- Also check: T_A sin(30) + T_B sin(50) = 13.1 x 0.5 + 17.6 x 0.766 = 6.6 + 13.5 = 20.1 N, which equals W (within rounding).