5) A person on a motor cycle gains a speed of 10kmph to 60kmph in 10 secs. The angular rotation of the needle in the speedometer is about 200 degrees. What will be the angular speed in degrees/sec?
The angular speed of the speedomenter needle while accelerating at that rate is:
w = (200/360)*(2 pi)rad/10 sec
= (5/9)*2*pi/10 = pi/9 radians/s
Well, I could give you a technical answer, but I have a feeling you're looking for a joke instead. So, here it goes: Why did the motorcycle enroll in math class? Because it wanted to learn about angular speed and make its speedometer needle spin faster!
To find the angular speed in degrees per second, we need to determine the angular displacement and divide it by the time taken.
1) Convert the speed from km/h to m/s:
- The initial speed is 10 km/h, which is equal to 10 × (1000/3600) m/s.
- The final speed is 60 km/h, which is equal to 60 × (1000/3600) m/s.
2) Calculate the average speed:
- The average speed is the total distance traveled divided by the total time taken.
- The distance traveled is the average speed multiplied by the time taken:
Distance = Average speed × Time
Distance = (Initial speed + Final speed) / 2 × Time
Distance = [(10 × (1000/3600)) + (60 × (1000/3600))] / 2 × 10
Distance = [(10 × 1000) + (60 × 1000)] / (3600 × 2) × 10
Distance = (10,000 + 60,000) / 72,000 × 10
Distance = 70,000 / 72,000 × 10
Distance ≈ 9.722 m
3) Convert the distance traveled into angular displacement:
- The angular displacement is given as 200 degrees.
- 360 degrees is equivalent to one complete revolution of the needle.
- So, the angular displacement is 200/360 of a complete revolution.
4) Calculate the angular speed:
- Angular speed = Angular displacement / Time
- Angular speed = (200/360) / 10
- Angular speed = (2/36) / 10
- Angular speed = 2/360
- Angular speed = 1/180 rad/s
5) Convert the angular speed from radians per second to degrees per second:
- 1 revolution = 360 degrees
- 2π radians = 1 revolution
- 1 radian = 360/2π degrees
- Angular speed (in degrees per second) = (1/180) × (360/2π)
- Angular speed ≈ 1.91 degrees per second
Therefore, the angular speed of the needle on the speedometer is approximately 1.91 degrees per second.
To find the angular speed in degrees/sec, we need to determine the change in angle over the given time period. This can be done using the formula:
Angular speed (in degrees/sec) = Change in angle (in degrees) / Time taken (in seconds)
In this case, we are given that the angular rotation of the needle in the speedometer is about 200 degrees and the time taken is 10 seconds. Plugging these values into the formula, we get:
Angular speed (in degrees/sec) = 200 degrees / 10 seconds
Simplifying further, we find:
Angular speed (in degrees/sec) = 20 degrees/sec
Therefore, the angular speed of the needle in the speedometer is 20 degrees/sec.