An object with a mass of 25 kg hangs from the ceiling. It is pulled sideways until the string forms an angle of 30° with the ceiling. Use a scale drawing or calculation and determi... An object with a mass of 25 kg hangs from the ceiling. It is pulled sideways until the string forms an angle of 30° with the ceiling. Use a scale drawing or calculation and determine the tension in the string as well as the applied force with which it is pulled sideways. Read more

Steps to Solve

1. Identify the Forces and Draw a Diagram

   The object has a mass of $25 , \text{kg}$. The gravitational force (weight) acting on it can be calculated using the equation:

   $$ F_g = m \cdot g $$

   where $g$ is the acceleration due to gravity ($9.81 , \text{m/s}^2$).

2. Calculate the Weight (Gravitational Force)

   Plugging in the values we have:

   $$ F_g = 25 , \text{kg} \cdot 9.81 , \text{m/s}^2 = 245.25 , \text{N} $$

   This is the downward force acting on the object.

3. Resolve the Tension in the String

   When the string is pulled to form a $30^\circ$ angle with the ceiling, the tension ($T$) in the string can be resolved into two components:

   • Vertical component: $T_y = T \cdot \cos(30^\circ)$
   • Horizontal component: $T_x = T \cdot \sin(30^\circ)$

4. Set Up the Equation for Vertical Forces

   In equilibrium, the vertical component of tension must balance the weight of the object:

   $$ T_y = F_g $$

   Therefore,

   $$ T \cdot \cos(30^\circ) = 245.25 , \text{N} $$

5. Calculate Tension in the String

   From the previous equation, solve for tension $T$:

   $$ T = \frac{245.25 , \text{N}}{\cos(30^\circ)} $$

   Use the value $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$.

6. Final Calculation of Tension

   Substitute the values:

   $$ T = \frac{245.25 , \text{N}}{0.866} \approx 283.67 , \text{N} $$