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As seen from the side view, what is the direction of the linear acceleration, if any, of point P, at time t = 0? Justify your answer. Explain how the student could use the function for 0(t) to determine the value of the angular speed w and the angular acceleration a after the roller has undergone a single rotation. In what direction is the acceleration of point P at the instant the roller has completed a single rotation? Justify your answer.

As seen from the side view, what is the direction of the linear acceleration, if any, of point P, at time t = 0? Justify your answer. Explain how the student could use the function for 0(t) to determine the value of the angular speed w and the angular acceleration a after the roller has undergone a single rotation. In what direction is the acceleration of point P at the instant the roller has completed a single rotation? Justify your answer.

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### Understanding Rotational Motion of a Roller with a Thin Metal Sheet #### Diagram Explanation: The diagram displays a long, thin sheet of flexible metal wrapped multiple times around a plastic roller with a radius \( R \). There are two views provided: - **Perspective View**: This view shows the setup from an angle, emphasizing how the metal sheet hangs vertically from the roller's edge. - **Side View**: This view looks directly at the side of the roller, highlighting the marking \( P \) on the roller at a distance \( r \) from the roller's center. #### Problem Description: Initially, a small part of the metal sheet hangs vertically from the roller's edge, with the roller held stationary. As the roller is released at time \( t = 0 \), it rotates while the metal sheet unwinds. A student observes and records the angular displacement \( \theta \) of the roller over a time interval \( T \). During this period, it is assumed that the mass of the metal sheet on the roller changes negligibly, and frictional effects are minimal. The equation \(\theta(t) = At^4 + Bt^2\) (with positive constants \( A \) and \( B \)) approximates the roller's motion over time. #### Analysis and Questions: - **(a) Direction of Linear Acceleration at \( t = 0 \)** - **Question**: As seen from the side view, what is the direction of the linear acceleration, if any, of point \( P \) at time \( t = 0 \)? Justify your answer. - **Hint**: Consider the initial tension in the metal sheet and the resulting force applying on point \( P \). - **(b) Determining Angular Speed and Acceleration** - **Question**: Explain how the student could use the function for \( \theta(t) \) to determine the value of the angular speed \( \omega \) and the angular acceleration \( \alpha \) after the roller has undergone a single rotation. - **Hint**: Use derivatives of \( \theta(t) \) to find angular velocity (\( \omega = \frac{d\theta}{dt} \)) and angular acceleration (\( \alpha = \frac{d^2\theta}{dt^2} \)). - **(c) Acceleration of Point \( P \) After a Single Rotation** - **
Transcribed Image Text:### Understanding Rotational Motion of a Roller with a Thin Metal Sheet #### Diagram Explanation: The diagram displays a long, thin sheet of flexible metal wrapped multiple times around a plastic roller with a radius \( R \). There are two views provided: - **Perspective View**: This view shows the setup from an angle, emphasizing how the metal sheet hangs vertically from the roller's edge. - **Side View**: This view looks directly at the side of the roller, highlighting the marking \( P \) on the roller at a distance \( r \) from the roller's center. #### Problem Description: Initially, a small part of the metal sheet hangs vertically from the roller's edge, with the roller held stationary. As the roller is released at time \( t = 0 \), it rotates while the metal sheet unwinds. A student observes and records the angular displacement \( \theta \) of the roller over a time interval \( T \). During this period, it is assumed that the mass of the metal sheet on the roller changes negligibly, and frictional effects are minimal. The equation \(\theta(t) = At^4 + Bt^2\) (with positive constants \( A \) and \( B \)) approximates the roller's motion over time. #### Analysis and Questions: - **(a) Direction of Linear Acceleration at \( t = 0 \)** - **Question**: As seen from the side view, what is the direction of the linear acceleration, if any, of point \( P \) at time \( t = 0 \)? Justify your answer. - **Hint**: Consider the initial tension in the metal sheet and the resulting force applying on point \( P \). - **(b) Determining Angular Speed and Acceleration** - **Question**: Explain how the student could use the function for \( \theta(t) \) to determine the value of the angular speed \( \omega \) and the angular acceleration \( \alpha \) after the roller has undergone a single rotation. - **Hint**: Use derivatives of \( \theta(t) \) to find angular velocity (\( \omega = \frac{d\theta}{dt} \)) and angular acceleration (\( \alpha = \frac{d^2\theta}{dt^2} \)). - **(c) Acceleration of Point \( P \) After a Single Rotation** - **
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