Question
A stone attached at the end of the string of length 0.7m is whirled around a circular path. If it takes 0.1seconds to make 1 complete revolution.calculate
1) frequency of revolution
2) angular velocity
3) linear speed
3) centripetal acceleration
1) frequency of revolution
2) angular velocity
3) linear speed
3) centripetal acceleration
Answers
GPT-4o mini
Let's break down the problem and calculate each of the requested quantities step by step.
1. **Frequency of Revolution (f)**:
The frequency \( f \) is calculated as the number of revolutions per second. Since the stone takes 0.1 seconds to make 1 complete revolution, the frequency can be calculated using the formula:
\[
f = \frac{1}{T}
\]
where \( T \) is the period (time for one complete revolution).
\[
T = 0.1 \text{ seconds} \Rightarrow f = \frac{1}{0.1} = 10 \text{ Hz}
\]
2. **Angular Velocity (ω)**:
The angular velocity \( \omega \) is related to the frequency by the following equation:
\[
\omega = 2\pi f
\]
Substituting the value of \( f \):
\[
\omega = 2\pi \times 10 \approx 62.83 \text{ rad/s}
\]
3. **Linear Speed (v)**:
The linear speed \( v \) can be calculated using the formula:
\[
v = r \omega
\]
where \( r \) is the radius (length of the string). Given \( r = 0.7 \) m:
\[
v = 0.7 \times 62.83 \approx 43.98 \text{ m/s}
\]
4. **Centripetal Acceleration (a_c)**:
The centripetal acceleration \( a_c \) can be calculated using the formula:
\[
a_c = \frac{v^2}{r}
\]
Substituting the values calculated:
\[
a_c = \frac{(43.98)^2}{0.7} \approx \frac{1936.80}{0.7} \approx 2767.71 \text{ m/s}^2
\]
### Summary of Results:
1. Frequency of Revolution: \( 10 \text{ Hz} \)
2. Angular Velocity: \( \approx 62.83 \text{ rad/s} \)
3. Linear Speed: \( \approx 43.98 \text{ m/s} \)
4. Centripetal Acceleration: \( \approx 2767.71 \text{ m/s}^2 \)
1. **Frequency of Revolution (f)**:
The frequency \( f \) is calculated as the number of revolutions per second. Since the stone takes 0.1 seconds to make 1 complete revolution, the frequency can be calculated using the formula:
\[
f = \frac{1}{T}
\]
where \( T \) is the period (time for one complete revolution).
\[
T = 0.1 \text{ seconds} \Rightarrow f = \frac{1}{0.1} = 10 \text{ Hz}
\]
2. **Angular Velocity (ω)**:
The angular velocity \( \omega \) is related to the frequency by the following equation:
\[
\omega = 2\pi f
\]
Substituting the value of \( f \):
\[
\omega = 2\pi \times 10 \approx 62.83 \text{ rad/s}
\]
3. **Linear Speed (v)**:
The linear speed \( v \) can be calculated using the formula:
\[
v = r \omega
\]
where \( r \) is the radius (length of the string). Given \( r = 0.7 \) m:
\[
v = 0.7 \times 62.83 \approx 43.98 \text{ m/s}
\]
4. **Centripetal Acceleration (a_c)**:
The centripetal acceleration \( a_c \) can be calculated using the formula:
\[
a_c = \frac{v^2}{r}
\]
Substituting the values calculated:
\[
a_c = \frac{(43.98)^2}{0.7} \approx \frac{1936.80}{0.7} \approx 2767.71 \text{ m/s}^2
\]
### Summary of Results:
1. Frequency of Revolution: \( 10 \text{ Hz} \)
2. Angular Velocity: \( \approx 62.83 \text{ rad/s} \)
3. Linear Speed: \( \approx 43.98 \text{ m/s} \)
4. Centripetal Acceleration: \( \approx 2767.71 \text{ m/s}^2 \)