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Learning Goal: To understand two different techniques for computing the torque on an object due to an applied force. Imagine an object with a pivot point p at the origin of the coordinate system shown (Figure 1). The force vector F lies in the xy plane, and this force of magnitude Facts on the object at a point in the xy plane. The vector 7 is the position vector relative to the pivot point p to the point where F is applied. The torque on the object due to the force F is equal to the cross product 7=7 x F. When, as in this problem, the force vector and lever arm both lie in the xy plane of the paper or computer screen. only the z component of to Tangential component of the force Part A (Figure 2) Decompose the force vector F into radial (i.e., parallel to 7) and tangential (perpendicular to 7) components as shown. Find the magnitude of the radial tangential components, F, and F. You may assume that is between zero degrees. Enter your answer as an ordered pair. Express F and F, in terms of Fa ► View Available Hint(s)

Learning Goal: To understand two different techniques for computing the torque on an object due to an applied force. Imagine an object with a pivot point p at the origin of the coordinate system shown (Figure 1). The force vector F lies in the xy plane, and this force of magnitude Facts on the object at a point in the xy plane. The vector 7 is the position vector relative to the pivot point p to the point where F is applied. The torque on the object due to the force F is equal to the cross product 7=7 x F. When, as in this problem, the force vector and lever arm both lie in the xy plane of the paper or computer screen. only the z component of to Tangential component of the force Part A (Figure 2) Decompose the force vector F into radial (i.e., parallel to 7) and tangential (perpendicular to 7) components as shown. Find the magnitude of the radial tangential components, F, and F. You may assume that is between zero degrees. Enter your answer as an ordered pair. Express F and F, in terms of Fa ► View Available Hint(s)

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The image appears to be from an educational website focused on physics, specifically discussing the concept of torque and the moment arm of a force. **Text:** In this problem, you can use the Component of the Force method or the Moment Arm of the Force method. Note that in this problem, the positive z direction is perpendicular to the computer screen and points toward you (given by the right-hand rule \( \hat{i} \times \hat{j} = \hat{k} \)), so a positive torque would cause counterclockwise rotation about the z axis. **Figure:** - The figure shows a two-dimensional plane with an x-axis and y-axis. - A force vector \(\vec{F}\) is shown in red, angled and extending from the origin. - A dashed line represents the perpendicular distance from the pivot point to the force vector, illustrating the moment arm. **Educational Context:** - **Part B, Part C, Part D, Part E**: These sections likely provide step-by-step guidance or questions related to calculating torque using different methods. - **Moment Arm of the Force Explanation**: - The moment arm (shown as the dashed line) is the perpendicular distance from the axis of rotation (pivot point) to the line of action of the force vector \(\vec{F}\). - **Part F, Part G**: These sections are possibly follow-ups or additional exercises to reinforce understanding of the topic. This educational content is copyrighted by Pearson Education Inc., 2023. For further study, students are invited to use the submission and feedback features provided on the website.
Transcribed Image Text:The image appears to be from an educational website focused on physics, specifically discussing the concept of torque and the moment arm of a force. **Text:** In this problem, you can use the Component of the Force method or the Moment Arm of the Force method. Note that in this problem, the positive z direction is perpendicular to the computer screen and points toward you (given by the right-hand rule \( \hat{i} \times \hat{j} = \hat{k} \)), so a positive torque would cause counterclockwise rotation about the z axis. **Figure:** - The figure shows a two-dimensional plane with an x-axis and y-axis. - A force vector \(\vec{F}\) is shown in red, angled and extending from the origin. - A dashed line represents the perpendicular distance from the pivot point to the force vector, illustrating the moment arm. **Educational Context:** - **Part B, Part C, Part D, Part E**: These sections likely provide step-by-step guidance or questions related to calculating torque using different methods. - **Moment Arm of the Force Explanation**: - The moment arm (shown as the dashed line) is the perpendicular distance from the axis of rotation (pivot point) to the line of action of the force vector \(\vec{F}\). - **Part F, Part G**: These sections are possibly follow-ups or additional exercises to reinforce understanding of the topic. This educational content is copyrighted by Pearson Education Inc., 2023. For further study, students are invited to use the submission and feedback features provided on the website.
**Learning Goal:** To understand two different techniques for computing the torque on an object due to an applied force. Imagine an object with a pivot point \( P \) at the origin of the coordinate system shown (Figure 1). The force vector \( \vec{F} \) lies in the xy plane, and this force of magnitude \( F \) acts on the object at a point in the xy plane. The vector \( \vec{r} \) is the position vector relative to the pivot point \( P \) to the point where \( \vec{F} \) is applied. The torque on the object due to the force \( \vec{F} \) is equal to the cross product \( \vec{\tau} = \vec{r} \times \vec{F} \). When, as in this problem, the force vector and lever arm both lie in the xy plane of the paper or computer screen, only the z component of torque is nonzero. When the torque vector is parallel to the z axis (\( \vec{\tau} = \tau \hat{k} \)), it is easiest to find the magnitude and sign of the torque, \( \tau \), in terms of the angle \( \theta \) between the position and force vectors using one of two simple methods: the Tangential Component of the Force method or the Moment Arm of the Force method. In this problem, the positive z direction is perpendicular to the computer screen and points toward you (given by the right-hand rule: \( \hat{i} \times \hat{j} = \hat{k} \)), so a positive torque would cause counterclockwise rotation about the z axis. **Tangential Component of the Force** **Part A** (See Figure 2) Decompose the force vector \( \vec{F} \) into radial (i.e., parallel to \( \vec{r} \)) and tangential (perpendicular to \( \vec{r} \)) components as shown. Find the magnitude of the radial and tangential components, \( F_r \) and \( F_t \). You may assume that \( \theta \) is between zero and 90 degrees. Enter your answer as an ordered pair. Express \( F_r \) and \( F_t \) in terms of \( F \) and \( \theta \). Submit your answer in format: \( (F_r, F_t) \
Transcribed Image Text:**Learning Goal:** To understand two different techniques for computing the torque on an object due to an applied force. Imagine an object with a pivot point \( P \) at the origin of the coordinate system shown (Figure 1). The force vector \( \vec{F} \) lies in the xy plane, and this force of magnitude \( F \) acts on the object at a point in the xy plane. The vector \( \vec{r} \) is the position vector relative to the pivot point \( P \) to the point where \( \vec{F} \) is applied. The torque on the object due to the force \( \vec{F} \) is equal to the cross product \( \vec{\tau} = \vec{r} \times \vec{F} \). When, as in this problem, the force vector and lever arm both lie in the xy plane of the paper or computer screen, only the z component of torque is nonzero. When the torque vector is parallel to the z axis (\( \vec{\tau} = \tau \hat{k} \)), it is easiest to find the magnitude and sign of the torque, \( \tau \), in terms of the angle \( \theta \) between the position and force vectors using one of two simple methods: the Tangential Component of the Force method or the Moment Arm of the Force method. In this problem, the positive z direction is perpendicular to the computer screen and points toward you (given by the right-hand rule: \( \hat{i} \times \hat{j} = \hat{k} \)), so a positive torque would cause counterclockwise rotation about the z axis. **Tangential Component of the Force** **Part A** (See Figure 2) Decompose the force vector \( \vec{F} \) into radial (i.e., parallel to \( \vec{r} \)) and tangential (perpendicular to \( \vec{r} \)) components as shown. Find the magnitude of the radial and tangential components, \( F_r \) and \( F_t \). You may assume that \( \theta \) is between zero and 90 degrees. Enter your answer as an ordered pair. Express \( F_r \) and \( F_t \) in terms of \( F \) and \( \theta \). Submit your answer in format: \( (F_r, F_t) \
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