An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.
The centripetal acceleration of the aircraft is approximately 6.4 times the acceleration due to gravity.
Given:
- Radius, R = 1.00 km = 1000 m
- Speed, v = 900 km/h = 250 m/s
Step 1: Convert speed to SI units 900 km/h = 900 × (1000/3600) = 250 m/s
Step 2: Calculate centripetal acceleration Using the formula from the context: a_c = v²/R a_c = (250)²/1000 a_c = 62500/1000 a_c = 62.5 m/s²
Step 3: Compare with acceleration due to gravity Taking g = 9.8 m/s² Ratio = a_c/g = 62.5/9.8 6.4
Thus, the centripetal acceleration is about 6.4g, meaning it is 6.4 times the acceleration due to gravity.
Explanation
This question tests the application of centripetal acceleration formula a_c = v²/R from the chapter. The context clearly states that for an object moving with speed v in a circle of radius R, the magnitude of centripetal acceleration is v²/R. Students must remember to convert units to SI before calculation - km to m and km/h to m/s. The comparison with g is done by finding the ratio a_c/g.
Solution Steps
Step 1: Convert radius R = 1.00 km = 1000 m and speed v = 900 km/h = 250 m/s
Step 2: Apply centripetal acceleration formula a_c = v²/R = (250)²/1000 = 62.5 m/s²
Step 3: Calculate ratio a_c/g = 62.5/9.8 6.4