Linear and angular speed
Find w for each of the following:
1. the minute hand of a clock
2. the second hand of a clock
The answer to #1 should be pi/30, but I do not understand how to solve these.
To find the angular speed (w) of an object, you need to know the time it takes for the object to complete one full revolution (in radians) and divide it by the time taken.
1. For the minute hand of a clock:
Since the minute hand of a clock completes one full revolution in 60 minutes (1 hour), we can calculate the angular speed as follows:
w = (2π radians) / (60 minutes)
= π/30 radians per minute (as you correctly mentioned)
2. For the second hand of a clock:
The second hand of a clock completes one full revolution in 60 seconds (1 minute). Therefore, the angular speed can be calculated as follows:
w = (2π radians) / (60 seconds)
= π/30 radians per second
Both answers are correct:
- The angular speed of the minute hand of a clock is π/30 radians per minute.
- The angular speed of the second hand of a clock is π/30 radians per second.
To find the linear and angular speed of an object, we need to consider its distance traveled and the time taken to travel that distance.
Let's start by understanding the concepts of linear speed and angular speed:
1. Linear Speed: This refers to the rate at which an object moves along a straight path. It is typically measured in units such as meters per second (m/s) or kilometers per hour (km/h). Linear speed can be calculated by dividing the distance traveled by the time taken:
Linear Speed = Distance / Time
2. Angular Speed: This is the rate at which an object rotates or moves around a fixed point. It is measured in radians per second (rad/s) or degrees per second (°/s). Angular speed can be calculated by dividing the angle covered by an object by the time taken to cover that angle:
Angular Speed = Angle / Time
Now, let's apply these concepts to solve the given questions.
1. Minute Hand of a Clock:
The minute hand of a clock completes a full revolution (360 degrees or 2π radians) in 60 minutes (1 hour). To find the angular speed, we can divide the angle covered by the minute hand by the time taken.
Angle covered by the minute hand = 2π radians
Time taken = 60 minutes = 60 x 60 seconds
Angular Speed = 2π radians / (60 x 60 seconds)
Angular Speed = π/180 rad/s (simplifying the expression)
Note: The angular speed is often expressed in degrees per second, so we can multiply the above result by 180/π to convert radians to degrees:
Angular Speed = π/180 rad/s * (180/π °/rad)
Angular Speed = 1°/s
Therefore, the angular speed of the minute hand of a clock is 1°/s.
2. Second Hand of a Clock:
Similarly, the second hand of a clock completes a full revolution (360 degrees or 2π radians) in 60 seconds (1 minute). To find the angular speed, we can divide the angle covered by the second hand by the time taken.
Angle covered by the second hand = 2π radians
Time taken = 60 seconds
Angular Speed = 2π radians / 60 seconds
Angular Speed = π/30 rad/s (simplifying the expression)
Again, if we want to express the angular speed in degrees per second, we can convert radians to degrees:
Angular Speed = π/30 rad/s * (180/π °/rad)
Angular Speed = 6°/s
Therefore, the angular speed of the second hand of a clock is 6°/s.
I hope this explanation clarifies how to calculate the angular speed for the minute and second hands of a clock.