Consider a slider, modeled as a particle P of mass m, moving along a frictionless shaft. The slider moves in the horizontal plane (there is no gravity). At the instant shown, the speed of the slider is v, and the spring is stretched to length r > lo, where lo is its unstretched length. The spring forms an angle as shown and the spring constant k is unknown. If, at the instant shown, 0 = 0 and i = 0 in the polar frame then what must the spring constant be? Also determine the normal force at this instant. Express both in terms of one or more of the following variables: (v, m,r,lo,0). (Hint: first solve for r and è from the velocity kinematics.)

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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The problem involves a slider, modeled as a particle \( P \) with mass \( m \), moving along a frictionless shaft in the horizontal plane, where there is no gravity. At the indicated instant, the slider's speed is \( v \), and the spring is stretched to a length \( r \), greater than its unstretched length \( l_0 \). The spring forms an angle \( \theta \) as shown, and the spring constant \( k \) is unknown. Given that the angular acceleration \(\ddot{\theta} = 0\) and the radial acceleration \(\dot{r} = 0\) in the polar frame, the task is to determine the necessary spring constant. It also requires determining the normal force at this instant. These should be expressed in terms of one or more of the following variables: \( (v, m, r, l_0, \theta) \).

**Hint:** Begin by solving for \(\dot{r}\) and \(\dot{\theta}\) from the velocity kinematics.

**Diagram Explanation:**

- The diagram illustrates the slider \( P \) moving on a vertical shaft. The slider, connected to a spring, moves in the horizontal plane.
- The spring extends to a distance \( r \) from a fixed point \( O \), forming an angle \( \theta \) relative to the horizontal axis.
- Motion is indicated by vectors: \( v \) for velocity of the slider and \( \theta \) marking the angular direction formed by the spring. 

This educational setup involves deriving equations of motion and forces considering the physical constraints and is suitable for studies in mechanics and dynamics.
Transcribed Image Text:The problem involves a slider, modeled as a particle \( P \) with mass \( m \), moving along a frictionless shaft in the horizontal plane, where there is no gravity. At the indicated instant, the slider's speed is \( v \), and the spring is stretched to a length \( r \), greater than its unstretched length \( l_0 \). The spring forms an angle \( \theta \) as shown, and the spring constant \( k \) is unknown. Given that the angular acceleration \(\ddot{\theta} = 0\) and the radial acceleration \(\dot{r} = 0\) in the polar frame, the task is to determine the necessary spring constant. It also requires determining the normal force at this instant. These should be expressed in terms of one or more of the following variables: \( (v, m, r, l_0, \theta) \). **Hint:** Begin by solving for \(\dot{r}\) and \(\dot{\theta}\) from the velocity kinematics. **Diagram Explanation:** - The diagram illustrates the slider \( P \) moving on a vertical shaft. The slider, connected to a spring, moves in the horizontal plane. - The spring extends to a distance \( r \) from a fixed point \( O \), forming an angle \( \theta \) relative to the horizontal axis. - Motion is indicated by vectors: \( v \) for velocity of the slider and \( \theta \) marking the angular direction formed by the spring. This educational setup involves deriving equations of motion and forces considering the physical constraints and is suitable for studies in mechanics and dynamics.
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