1. A 75.0 cm diameter high-speed flywheel in an electric motor is rotating when a power failure suddenly occurs. As a result, a point on the rim of the flywheel slows down such that its tangential velocity is given by: v, = 19.5 – 0.263t , where t is in s and v; is in m/s. (a) What is the initial angular velocity and total acceleration vector magnitude?
1. A 75.0 cm diameter high-speed flywheel in an electric motor is rotating when a power failure suddenly occurs. As a result, a point on the rim of the flywheel slows down such that its tangential velocity is given by: v, = 19.5 – 0.263t , where t is in s and v; is in m/s. (a) What is the initial angular velocity and total acceleration vector magnitude?
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Chapter1: Units, Trigonometry. And Vectors
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Transcribed Image Text:### Questions
(b) How long does it take for the flywheel to stop?
(c) Find the magnitude of the total acceleration at 72.0 s.
![### Problem Statement:
1. A 75.0 cm diameter high-speed flywheel in an electric motor is rotating when a power failure suddenly occurs. As a result, a point on the rim of the flywheel slows down such that its tangential velocity is given by:
\[ v_t = 19.5 - 0.263t \]
where \( t \) is in seconds and \( v_t \) is in meters per second.
(a) What is the initial angular velocity and total acceleration vector magnitude?
### Diagram:
- The image contains a diagram of a flywheel with a center indicated by a point and two perpendicular arrows emanating from it, representing rotational motion and axes.
- The flywheel is depicted as having concentric circles, denoting different parts or sections of the wheel.
### Solution Approach:
1. **Initial Angular Velocity (\(\omega_0\))**:
- The tangential velocity \( v_t \) is related to the angular velocity by the equation \( v_t = r \cdot \omega \), where \( r \) is the radius of the flywheel.
- Given the diameter is 75.0 cm, the radius \( r \) is 0.375 m.
- Using \( v_t = 19.5 \) m/s (the initial velocity when \( t = 0 \)), solve for the initial angular velocity:
\[
\omega_0 = \frac{v_t}{r} = \frac{19.5}{0.375}
\]
2. **Total Acceleration**:
- Total acceleration consists of tangential acceleration (\( a_t \)) and radial (centripetal) acceleration (\( a_r \)).
- Tangential acceleration (\( a_t \)) is given by the derivative of the velocity function \( v_t(t) \). Thus,
\[
a_t = \frac{d}{dt}(19.5 - 0.263t) = -0.263 \, \text{m/s}^2
\]
- Radial acceleration (\( a_r \)) is given by \( a_r = \omega^2 \cdot r \). Use the initial angular velocity \( \omega_0 \) for initial radial acceleration:
\[
a_r = \omega_0^2 \cdot r
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Transcribed Image Text:### Problem Statement:
1. A 75.0 cm diameter high-speed flywheel in an electric motor is rotating when a power failure suddenly occurs. As a result, a point on the rim of the flywheel slows down such that its tangential velocity is given by:
\[ v_t = 19.5 - 0.263t \]
where \( t \) is in seconds and \( v_t \) is in meters per second.
(a) What is the initial angular velocity and total acceleration vector magnitude?
### Diagram:
- The image contains a diagram of a flywheel with a center indicated by a point and two perpendicular arrows emanating from it, representing rotational motion and axes.
- The flywheel is depicted as having concentric circles, denoting different parts or sections of the wheel.
### Solution Approach:
1. **Initial Angular Velocity (\(\omega_0\))**:
- The tangential velocity \( v_t \) is related to the angular velocity by the equation \( v_t = r \cdot \omega \), where \( r \) is the radius of the flywheel.
- Given the diameter is 75.0 cm, the radius \( r \) is 0.375 m.
- Using \( v_t = 19.5 \) m/s (the initial velocity when \( t = 0 \)), solve for the initial angular velocity:
\[
\omega_0 = \frac{v_t}{r} = \frac{19.5}{0.375}
\]
2. **Total Acceleration**:
- Total acceleration consists of tangential acceleration (\( a_t \)) and radial (centripetal) acceleration (\( a_r \)).
- Tangential acceleration (\( a_t \)) is given by the derivative of the velocity function \( v_t(t) \). Thus,
\[
a_t = \frac{d}{dt}(19.5 - 0.263t) = -0.263 \, \text{m/s}^2
\]
- Radial acceleration (\( a_r \)) is given by \( a_r = \omega^2 \cdot r \). Use the initial angular velocity \( \omega_0 \) for initial radial acceleration:
\[
a_r = \omega_0^2 \cdot r
\]
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