A particle of mass m is attached to a spring, as shown below. Initially, the spring is unstretched with length lo and at rest. The spring constant is k. An impulse is applied to give an initial ve- locity vo in the direction a. The motion occurs in the horizontal plane (ignore gravity and friction). Part A: What is the angular impulse about point O required to produce this speed and direc- tion? Part B: Show that the equations of motion for the position of particle P with respect to O (in a polar frame) are as follows: Ö // -2r0 1 -k(r-lo) m Ÿ +r0², where r is the distance of P with respect to O and is the corresponding angle measured counter- clockwise from the horizontal. (Hint: Define a polar reference frame and use Newton's Second Law (angular momentum form, Mp/o = (hp/o) to find and use Newton's Second Law (standard form, Fp = mpap/o) with the polar acceleration to find r.) Values: lo=1 m, mp = 2 kg, vo = 3 m/s, a = 45 deg. (1) (2) lo m No fa P
A particle of mass m is attached to a spring, as shown below. Initially, the spring is unstretched with length lo and at rest. The spring constant is k. An impulse is applied to give an initial ve- locity vo in the direction a. The motion occurs in the horizontal plane (ignore gravity and friction). Part A: What is the angular impulse about point O required to produce this speed and direc- tion? Part B: Show that the equations of motion for the position of particle P with respect to O (in a polar frame) are as follows: Ö // -2r0 1 -k(r-lo) m Ÿ +r0², where r is the distance of P with respect to O and is the corresponding angle measured counter- clockwise from the horizontal. (Hint: Define a polar reference frame and use Newton's Second Law (angular momentum form, Mp/o = (hp/o) to find and use Newton's Second Law (standard form, Fp = mpap/o) with the polar acceleration to find r.) Values: lo=1 m, mp = 2 kg, vo = 3 m/s, a = 45 deg. (1) (2) lo m No fa P
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![# Problem
A particle of mass \( m \) is attached to a spring, as shown below. Initially, the spring is unstretched with length \( l_0 \) and at rest. The spring constant is \( k \). An impulse is applied to give an initial velocity \( v_0 \) in the direction \( \alpha \). The motion occurs in the horizontal plane (ignore gravity and friction).
**Part A:** What is the angular impulse about point \( O \) required to produce this speed and direction?
**Part B:** Show that the equations of motion for the position of particle \( P \) with respect to \( O \) (in a polar frame) are as follows:
\[
\ddot{\theta} = \frac{-2\dot{r}\dot{\theta}}{r} \tag{1}
\]
\[
\ddot{r} = -\frac{k(r-l_0)}{m} + r\dot{\theta}^2 \tag{2}
\]
where \( r \) is the distance of \( P \) with respect to \( O \) and \( \theta \) is the corresponding angle measured counter-clockwise from the horizontal.
(Hint: Define a polar reference frame and use Newton’s Second Law (angular momentum form, \( M_{P/O} = -I \frac{d}{dt} (h_{P/O}) \) to find \( \ddot{\theta} \) and use Newton’s Second Law (standard form, \( F_P = m_pa_{P/O} \) with the polar acceleration to find \( \ddot{r} \).)
**Values:** \( l_0 = 1 \, \text{m}, \, m_P = 2 \, \text{kg}, \, v_0 = 3 \, \text{m/s}, \, \alpha = 45 \, \text{deg}. \)
## Diagram Explanation
The diagram shows a horizontal spring attached to a particle \( P \). The spring has an unstretched length \( l_0 \) and is initially at rest. An impulse gives the particle an initial velocity \( v_0 \) at an angle \( \alpha \) to the horizontal. The setup ignores gravity and friction, simulating horizontal plane motion.
The spring constant is labeled \( k \), the length of the spring is marked \( l_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd23cf25f-e0e1-420e-8dcb-8ea8662c7deb%2Fa2ed702b-3036-4f3a-b4ae-49df80ccf4c5%2Fi1jfrdk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Problem
A particle of mass \( m \) is attached to a spring, as shown below. Initially, the spring is unstretched with length \( l_0 \) and at rest. The spring constant is \( k \). An impulse is applied to give an initial velocity \( v_0 \) in the direction \( \alpha \). The motion occurs in the horizontal plane (ignore gravity and friction).
**Part A:** What is the angular impulse about point \( O \) required to produce this speed and direction?
**Part B:** Show that the equations of motion for the position of particle \( P \) with respect to \( O \) (in a polar frame) are as follows:
\[
\ddot{\theta} = \frac{-2\dot{r}\dot{\theta}}{r} \tag{1}
\]
\[
\ddot{r} = -\frac{k(r-l_0)}{m} + r\dot{\theta}^2 \tag{2}
\]
where \( r \) is the distance of \( P \) with respect to \( O \) and \( \theta \) is the corresponding angle measured counter-clockwise from the horizontal.
(Hint: Define a polar reference frame and use Newton’s Second Law (angular momentum form, \( M_{P/O} = -I \frac{d}{dt} (h_{P/O}) \) to find \( \ddot{\theta} \) and use Newton’s Second Law (standard form, \( F_P = m_pa_{P/O} \) with the polar acceleration to find \( \ddot{r} \).)
**Values:** \( l_0 = 1 \, \text{m}, \, m_P = 2 \, \text{kg}, \, v_0 = 3 \, \text{m/s}, \, \alpha = 45 \, \text{deg}. \)
## Diagram Explanation
The diagram shows a horizontal spring attached to a particle \( P \). The spring has an unstretched length \( l_0 \) and is initially at rest. An impulse gives the particle an initial velocity \( v_0 \) at an angle \( \alpha \) to the horizontal. The setup ignores gravity and friction, simulating horizontal plane motion.
The spring constant is labeled \( k \), the length of the spring is marked \( l_
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