**Educational Content on Circular Motion Dynamics****Scenario:**A puck of mass 0.570 kg is attached to the end of a cord 1.20 m long. The puck moves in a horizontal circle as shown in the figure. If the cord can withstand a maximum tension of 55.0 N, what is the maximum speed at which the puck can move before the cord breaks? Assume the string remains horizontal during the motion.*Diagram:*The diagram illustrates an overhead view of a puck moving in a circular path in a horizontal plane. A force, \(\vec{F}_T\), directed toward the center of the circle, keeps the puck moving in its circular path. The tension in the cord, also labeled \(\vec{F}_T\), has its direction indicated towards the circle's center.**Solution:****Conceptualize:**It makes sense that the stronger the cord, the faster the puck can move before the cord breaks. Also, we expect a more massive puck to break the cord at a higher speed. (Imagine whirling a bowling ball on the cord!)**Categorize:**Because the puck moves in a circular path, we model it as a particle in uniform circular motion.**Analyze:**(Use the following as necessary: \(T\), \(m\), \(v\), and \(r\). Do not substitute numerical values; use variables only.)Incorporate the tension and the centripetal acceleration into Newton’s second law:\[ T = m \frac{v^2}{r} \]**Solve for \(v\):**\[ \text{(1)} \quad v = \sqrt{\frac{T \cdot r}{m}} \]Find the maximum speed (in m/s) the puck can have, which corresponds to the maximum tension the string can withstand:\[ v_{\text{max}} = \sqrt{\frac{55.0 \, \text{N} \cdot 1.20 \, \text{m}}{0.570 \, \text{kg}}} \, \text{m/s} \]**Finalize:**Equation (1) shows that \(v\) increases with \(T\) and decreases with a larger \(m\), as we expected from our conceptualization of the problem.**Exercise:**Calculate the tension (in N) in the cord if the speed of the object is

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A puck of mass 0.570 kg is attached to the end of a cord 1.20 m long. The puck moves in a horizontal circle as shown in the figure. If the cord can withstand a maximum tension of 55.0ON, what is the maximum speed at which the puck can move before the cord breaks? Assume the string remains horizontal during the motion. A force F,, directed toward the center of the circle, keeps the puck moving in its circular path. Overhead view of a puck moving in a circular path in a horizontal plane. SOLUTION Conceptualize It makes sense that the stronger the cord, the faster the puck can move before the cord breaks. Also, we expect a more massive puck to break the cord at a -Select- speed. (Imagine whirling a bowling ball on the cord!) Categorize Because the puck moves in a circular path, we model it as a particle -Select- Analyze (Use the following as necessary: T, m, v, and r. Do not substitute numerical values; use variables only.) Incorporate the tension and the centripetal acceleration into Newton's second law: (1) v= Find the maximum speed (in m/s) the puck can have, which corresponds to the maximum tension the string can withstand: Vmax O m/s Finalize Equation (1) shows that v-Select-- with T and Select-- Vwith a larger m, as we expected from our conceptualization of the problem. EXERCISE Calculate the tension (in N) in the cord if the speed of the object is 3.0 m/s. Hint N

A puck of mass 0.570 kg is attached to the end of a cord 1.20 m long. The puck moves in a horizontal circle as shown in the figure. If the cord can withstand a maximum tension of 55.0ON, what is the maximum speed at which the puck can move before the cord breaks? Assume the string remains horizontal during the motion. A force F,, directed toward the center of the circle, keeps the puck moving in its circular path. Overhead view of a puck moving in a circular path in a horizontal plane. SOLUTION Conceptualize It makes sense that the stronger the cord, the faster the puck can move before the cord breaks. Also, we expect a more massive puck to break the cord at a -Select- speed. (Imagine whirling a bowling ball on the cord!) Categorize Because the puck moves in a circular path, we model it as a particle -Select- Analyze (Use the following as necessary: T, m, v, and r. Do not substitute numerical values; use variables only.) Incorporate the tension and the centripetal acceleration into Newton's second law: (1) v= Find the maximum speed (in m/s) the puck can have, which corresponds to the maximum tension the string can withstand: Vmax O m/s Finalize Equation (1) shows that v-Select-- with T and Select-- Vwith a larger m, as we expected from our conceptualization of the problem. EXERCISE Calculate the tension (in N) in the cord if the speed of the object is 3.0 m/s. Hint N

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