A group of students observes that a wooden block (m = 0.40 kg) on the end of a string with a radius of 0.7 meters makes 9 rotations in 20.7 seconds when twirled.
a)What is the block's tangential (linear) speed?
b)What is the block's angular speed?
a) Well, to calculate the block's tangential speed, we need to know the distance it travels in a given time. In this case, we have the number of rotations and the radius of the string. So, the tangential speed is like a secret agent disguised as the circumference of a circle.
The formula for the circumference of a circle is C = 2πr. In this case, the radius is 0.7 meters. So, the secret agent tangential speed is given by multiplying the circumference by the number of rotations.
Tangential Speed = (2πr)(Number of Rotations)
Therefore, Tangential Speed = (2π)(0.7)(9 rotations)
Now, let's unleash our secret agent and do the math to get the answer!
b) But wait, there's more! We can also calculate the block's angular speed. Angular speed is like a whirlwind, ready to blow your mind with its speed. It is measured in radians per second.
To calculate angular speed, we need to divide the number of rotations by the time it takes. So, the angular speed is given by the formula:
Angular Speed = (Number of Rotations) / (Time)
Angular Speed = 9 rotations / 20.7 seconds
Now, let's put on our superhero capes and calculate the angular speed!
Voila! We have the block's tangential (linear) speed and angular speed all measured up!
To find the block's tangential (linear) speed, you can use the formula:
Tangential speed = 2 * π * radius / time
Where:
- π (pi) is a mathematical constant approximately equal to 3.14,
- radius is the distance from the center of rotation to the block's position,
- time is the time taken to complete the rotations.
Given:
radius = 0.7 meters
time = 20.7 seconds
a) To find the block's tangential speed:
Tangential speed = 2 * π * 0.7 meters / 20.7 seconds
Let's calculate it:
Tangential speed = 2 * 3.14 * 0.7 meters / 20.7 seconds
Tangential speed = 1.4 * 3.14 * 0.7 meters / 20.7 seconds
Tangential speed ≈ 2.755 meters/second
Therefore, the block's tangential (linear) speed is approximately 2.755 meters/second.
To find the block's angular speed, you can use the formula:
Angular speed = 2 * π * rotations / time
Where:
- π (pi) is a mathematical constant approximately equal to 3.14,
- rotations is the number of complete rotations made by the block,
- time is the time taken to complete the rotations.
Given:
rotations = 9
time = 20.7 seconds
b) To find the block's angular speed:
Angular speed = 2 * π * 9 rotations / 20.7 seconds
Let's calculate it:
Angular speed = 2 * 3.14 * 9 rotations / 20.7 seconds
Angular speed = 18 * 3.14 rotations / 20.7 seconds
Angular speed ≈ 2.748 rotations/second
Therefore, the block's angular speed is approximately 2.748 rotations/second.
To find the block's tangential (linear) speed and angular speed, we can use the given information about the number of rotations and the time it takes to complete those rotations.
a) Tangential (linear) speed is the speed at which an object moves along its circular path. It can be calculated using the formula:
Tangential speed = (2πr * N) / t
Where:
- r is the radius of the circular path (0.7 meters)
- N is the number of rotations (9 rotations)
- t is the time taken to complete those rotations (20.7 seconds)
Substituting the values into the formula:
Tangential speed = (2π * 0.7 * 9) / 20.7
Now, let's calculate the tangential speed:
Tangential speed = (12.6π) / 20.7
Tangential speed ≈ 1.92 meters per second
Therefore, the block's tangential speed is approximately 1.92 meters per second.
b) Angular speed is the rate at which an object rotates or moves angularly. It can be calculated using the formula:
Angular speed = (2π * N) / t
Where:
- N is the number of rotations (9 rotations)
- t is the time taken to complete those rotations (20.7 seconds)
Substituting the values into the formula:
Angular speed = (2π * 9) / 20.7
Now, let's calculate the angular speed:
Angular speed = (18π) / 20.7
Angular speed ≈ 2.75 radians per second
Therefore, the block's angular speed is approximately 2.75 radians per second.
a. v=2PIr/(9/20.7)
b. angular speed=v/r