1. A rope of 17 mm diameter is wound once around drum of I m diamet The one end of the rope is tied to a spring balance, and the other end to a mass of 50 kg . Engine speed is \( 500 \mathrm{r} / \mathrm{min} \). and the spring balance reading \( =30 \mathrm{~N} \). Calculate the brake power. ( 12 .

To calculate the brake power (BP) of the engine, we can use the formula: \[ \text{BP} = \frac{2 \pi N T}{60} \] where: - \( N \) is the speed of the engine in revolutions per minute (rpm), - \( T \) is the torque in Newton-meters (Nm). ### Step 1: Calculate the Torque (T) The torque can be calculated using the force measured by the spring balance and the radius of the drum. The force \( F \) is given as 30 N, and the radius \( r \) of the drum can be calculated from its diameter. Given: - Diameter of the drum = 1 m - Radius of the drum \( r = \frac{1}{2} \) m = 0.5 m The torque \( T \) is given by: \[ T = F \times r \] Substituting the values: \[ T = 30 \, \text{N} \times 0.5 \, \text{m} = 15 \, \text{Nm} \] ### Step 2: Calculate the Brake Power (BP) Now, substituting the values of \( N \) and \( T \) into the brake power formula: Given: - Engine speed \( N = 500 \, \text{rpm} \) - Torque \( T = 15 \, \text{Nm} \) \[ \text{BP} = \frac{2 \pi (500) (15)}{60} \] Calculating this step-by-step: 1. Calculate \( 2 \pi \): \[ 2 \pi \approx 6.2832 \] 2. Calculate \( 2 \pi N \): \[ 2 \pi N = 6.2832 \times 500 \approx 3141.6 \] 3. Now calculate \( \text{BP} \): \[ \text{BP} = \frac{3141.6 \times 15}{60} \] \[ \text{BP} = \frac{47124}{60} \approx 785.4 \, \text{W} \] ### Step 3: Convert to kW To convert watts to kilowatts: \[ \text{BP} \approx 0.785 \, \text{kW} \] ### Final Answer The brake power of the engine is approximately **0.785 kW**.