Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Educational Content: Newton’s Second Law of Motion **Problem Statement:** Use Newton’s Second Law of Motion to find the force required so that a particle of mass 2 has position \( \mathbf{r}(t) = \langle t^2, t^4, \sin(2t) \rangle \). **Select one:** a. \( \langle 2, 12t^2, -4 \sin(2t) \rangle \) b. \( \langle 4, 24t^2, -8 \sin(2t) \rangle \) c. 28 d. 24 e. \( \langle 2t, 4t^3, 2 \cos(2t) \rangle \) --- ### Explanation: This problem involves using Newton’s Second Law, \( \mathbf{F} = m \mathbf{a} \), where: - \( m \) is the mass of the particle. - \( \mathbf{a} \) is the acceleration, which is the second derivative of the position function \( \mathbf{r}(t) \). Calculate the second derivative \( \mathbf{r}''(t) \) to get acceleration, then multiply by the mass (2 in this case) to find the force. Choose the correct answer based on your calculation.
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Transcribed Image Text:### Educational Content: Newton’s Second Law of Motion **Problem Statement:** Use Newton’s Second Law of Motion to find the force required so that a particle of mass 2 has position \( \mathbf{r}(t) = \langle t^2, t^4, \sin(2t) \rangle \). **Select one:** a. \( \langle 2, 12t^2, -4 \sin(2t) \rangle \) b. \( \langle 4, 24t^2, -8 \sin(2t) \rangle \) c. 28 d. 24 e. \( \langle 2t, 4t^3, 2 \cos(2t) \rangle \) --- ### Explanation: This problem involves using Newton’s Second Law, \( \mathbf{F} = m \mathbf{a} \), where: - \( m \) is the mass of the particle. - \( \mathbf{a} \) is the acceleration, which is the second derivative of the position function \( \mathbf{r}(t) \). Calculate the second derivative \( \mathbf{r}''(t) \) to get acceleration, then multiply by the mass (2 in this case) to find the force. Choose the correct answer based on your calculation.
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