A small ball of mass m is suspended from a string of length L. The ball revolves with constant speed v in a horizontal circle of radius r as shown in the figure. (Because the string sweeps out the surface of a cone, the system is known as a conical pendulum.) Find an expression for v in terms of the geometry in the figure.Tcos eTsản emgmgA conical pendulum. The pathof the ball is a horizontalThe forces acting oncircle.the ball.SOLUTIONConceptualize Imagine the motion of the ball in figure (a) and convince yourself that the string sweeps out a cone and that the ball moves in a horizontal circle. What are the forces acting on the ball? (Select all that apply.)OfO mgORCategorize The ball in the figure does not accelerate vertically. Therefore, we model it as a particle in equilibrium in the vertical direction. It experiences a centripetal acceleration in the horizontal direction, so it is modeled as a particle --Select-Vin this direction.Analyze Let e represent the angle between the string and the vertical. In the diagram of forces acting on the ball in figure (b), the force Texerted by the string on the ball is resolved into a vertical component T cos(0) and a horizontal component T sin(0) actingSelect---the circular path.Apply the particle in equilibrium model in the vertical direction (Use the following as necessary: m and g.):= T cos(0) - mg = 0(1) T cos(0) =Use the equation F = ma = mfrom the particleuniform circular motion model in the horizontal direction:(2) F =T sin 0 = ma = mvsin(e)Divide Equation (2) by Equation (1) and use= tan(0) (Use the following as necessary: v, r, and g.):cos(0)tan(0) =Solve for v:V =V rg tan eIncorporate r =L sin(0) from the geometry in figure (a) (Use the following as necessary: L, g and 0):V =Finalize Notice that the speed is independent of the mass of the ball. Consider what happens when 0 goes to 90° so that the string is horizontal. Because the tangent of 90° is --Select-Vthe speed v is infinite, which tells us the string cannot possibly be horizontal. If it were, there would be no vertical component of the force T to balance the gravitational force on the ball.EXERCISEFor the conical pendulum described above, determine the following if m = 25.0 kg and 0 = 21°.Hint(a) the horizontal and vertical components of the force (in N) exerted by the string on the objectFH(b) the radial acceleration (in m/s) of the object (Enter the magnitude.)1m/s?

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figure. (Because the string sweeps out the surface of a cone, the system is known as a or v in terms of the geometry in the figure A conical pendulum. The path of the balsahora forces actine rde the bal SOLUTION Conceptualize Imagine the motion of the ball in figure (a) and convince yoursef that the string sweeps out a cone and that the ball moves in a horizontal cirde. What are the forces acting on the bal? (Select all that apply) Of Categorize The ball in the figure does not accelerate vertically. Therefore, we model it as a particle in equilibrium in the vertical direction. It experiences a centripetal acceleration in the horiazontal direction, se it is modeled as a particle-Selec- In this direction Analyze Let represent the angle between the string and the vertical. In the diagram of forces acting on the ball in figure (b), the force Texerted by the string on the bal is resolved into a vertical component Tcos and a horizontal component Tsin) actingSeed- Ve grolar path Apply the partidle in equilibrium model in the vertical direction (Use the following as necessary: mand g.): (1) Teos - Use the equation . from the particle in uniform circular motion model in the horizontal direction: (2) E.-Tn ma̟ - Divide Equation (2) by Equation (1) and use - tan (Use the following as necessary: v,r, and g.): tan) Solve for y V tane IncorporaterL sin( from the geometry in figure (a) (Use the following as necessary: Lg and Finalize Notice that the speed is independent of the mass of the ball. Consider what happens when goes to 90 so that the string is horizontal. Because the tangent of 90 is ESeled-V. the speed v is infinite, which tells us the string cannot possibly be horizontal. were, there would be no vertical component of the force Y to balance the gravitational force on the bal EXERCISE For the conical pendulum described above, determine the following if m - 25.0 kg ande- 21. Hint (a) the horizontal and vertical components of the force (in N) exerted by the string on the object

figure. (Because the string sweeps out the surface of a cone, the system is known as a or v in terms of the geometry in the figure A conical pendulum. The path of the balsahora forces actine rde the bal SOLUTION Conceptualize Imagine the motion of the ball in figure (a) and convince yoursef that the string sweeps out a cone and that the ball moves in a horizontal cirde. What are the forces acting on the bal? (Select all that apply) Of Categorize The ball in the figure does not accelerate vertically. Therefore, we model it as a particle in equilibrium in the vertical direction. It experiences a centripetal acceleration in the horiazontal direction, se it is modeled as a particle-Selec- In this direction Analyze Let represent the angle between the string and the vertical. In the diagram of forces acting on the ball in figure (b), the force Texerted by the string on the bal is resolved into a vertical component Tcos and a horizontal component Tsin) actingSeed- Ve grolar path Apply the partidle in equilibrium model in the vertical direction (Use the following as necessary: mand g.): (1) Teos - Use the equation . from the particle in uniform circular motion model in the horizontal direction: (2) E.-Tn ma̟ - Divide Equation (2) by Equation (1) and use - tan (Use the following as necessary: v,r, and g.): tan) Solve for y V tane IncorporaterL sin( from the geometry in figure (a) (Use the following as necessary: Lg and Finalize Notice that the speed is independent of the mass of the ball. Consider what happens when goes to 90 so that the string is horizontal. Because the tangent of 90 is ESeled-V. the speed v is infinite, which tells us the string cannot possibly be horizontal. were, there would be no vertical component of the force Y to balance the gravitational force on the bal EXERCISE For the conical pendulum described above, determine the following if m - 25.0 kg ande- 21. Hint (a) the horizontal and vertical components of the force (in N) exerted by the string on the object