An object attached to a 0.5 m string completes 4 rotations in one second. find the linear velocity(m/s) and the angular velocity(rad/s).
2π radians in each circle, so the angular speed is 8π rad/s
2π*0.5 = π meters in each circle, so the linear speed is 4π m/s
B
To find the linear velocity (v) and angular velocity (ω), we can use the formulas:
v = r × ω
ω = 2π × f
where:
- v is the linear velocity,
- r is the radius of the circular path (length of the string),
- ω is the angular velocity,
- f is the frequency of rotations per second,
- π is a mathematical constant.
Given:
- The string length (radius), r = 0.5 m
- The number of rotations per second, f = 4
First, let's find the angular velocity:
ω = 2π × f
= 2π × 4
= 8π rad/s
The angular velocity is 8π rad/s.
Next, let's find the linear velocity, using the formula:
v = r × ω
= 0.5 m × 8π rad/s
≈ 12.57 m/s
The linear velocity is approximately 12.57 m/s.
To find the linear velocity and angular velocity of an object attached to a string, we need to use the formulas related to circular motion.
First, let's define some variables:
- \(v\) = linear velocity of the object (in m/s)
- \(r\) = radius of the circular path (length of the string in this case, in meters)
- \(ω\) = angular velocity of the object (in rad/s)
- \(n\) = number of rotations completed by the object
- \(t\) = time taken to complete \(n\) rotations (in seconds)
We are given that the object completes 4 rotations in one second (\(t = 1\) s) and the length of the string (\(r\)) is 0.5 meters.
Linear velocity (\(v\)) can be calculated using the formula:
\[ v = \frac{2πr}{t} \]
Angular velocity (\(ω\)) can be calculated using the formula:
\[ ω = \frac{2πn}{t} \]
Substituting the given values, we can find the linear velocity and angular velocity:
\[ v = \frac{2π(0.5)}{1} \]
\[ ω = \frac{2π \cdot 4}{1} \]
Calculating these equations will give us the values of \(v\) and \(ω\).
Let's do the math:
\[ v = 3.14 \, \text{m/s} \]
\[ ω = 25.13 \, \text{rad/s} \]
Therefore, the linear velocity is 3.14 m/s and the angular velocity is 25.13 rad/s for an object rotating 4 times in 1 second on a 0.5 m string.