**Problem Context:**A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane, with the cord making a 30° angle with the vertical. **Objectives:**(a) Determine the ball’s speed.(b) If, instead, the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical?(c) If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at which the ball can move?**Diagram Overview:**- The diagram shows a hand holding a cord at an angle $\theta$ to the vertical.- The ball is revolving in a circular path, indicated by a dashed circle.**Considerations:**- The cord length is 1.5 m.- The angle with the vertical is initially 30°.- Use these parameters to solve the questions and understand the dynamics of circular motion.

The ball having a mass of 0.5 kg is revolved in a horizontal plane, tied to a 1.5 m long cord. While revolving, this cord makes an angle of 30 with the vertical. Using trigonometric relations, the radius of the horizontal circle traced by the ball for an angle of 30 with the vertical is given as r=lθl is the length of the cord=1.5 mθ=30°=0.5236 radr=1.50.5236r=2.8648 mAs the ball is tied to the cord, there will be a force of tension acting along the length of the cord. Let this force of tension be T, which acts along the length of the cord. This force of tension along the cord can be resolved into its horizontal and vertical components as, Tx=Tsinθ, horizontalTy=Tcosθ, vertical Also, the weight of the ball acts downwards vertically, and is thus balanced by the vertical component of tension. Thus, Ty=mgm is the mass of the ball and g is acceleration due to gravityTcosθ=0.5×9.8Tcos30=4.9T=4.90.866T=5.658 NAnd as the ball rotates in the horizontal circular plane, a centripetal force also acts on the ball, which is directed radially along the radius of the circular path of the ball, which is in the horizontal plane. This force is given as Fc=mv2rv is the linear velocity of the ball This centripetal force in the horizontal plane is then balanced by the horizontal component of tension Thus, Tx=FcTsinθ=mv2rv2=Trsinθmv=Trsinθmv=5.658×2.8648×sin300.5v=16.2v=4.026 m/s This is the required speed of the ballNow instead if the speed of the ball is 4 m/s, then the centripetal force acting on the ball will change, and will be Fc=mv2rFc=0.5×422.8648Fc=2.7925 N And since the centripetal force is balanced by the horizontal component of tension, Tx=FcTsinθ=2.7925sinθ=2.79255.658sinθ=0.49355θ=sin-1(0.49355)θ=29.5743°For a maximum tension force of 9.8 N, with an angle of 30, the horizontal tension component is Tsinθ=9.8×sin30Tsinθ=4.9 N And thus, Fc=TsinθFc=4.9mv2r=4.9v2=4.9×2.86480.5v2=28.07504v=28.07504v=5.2986 m/s This is the maximum speed with which the ball can revolveAnswers: a: v=4.026 m/s b: 29.5743o c: v=5.2986 m/s

Science

Physics

74. A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane, with the cord making a 30° angle with the verti- cal. (See Fig. P7.74.) (a) Determine the ball's speed. (b) If, instead, the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical? (c) If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at Figure P7.74 which the ball can move?

74. A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane, with the cord making a 30° angle with the verti- cal. (See Fig. P7.74.) (a) Determine the ball's speed. (b) If, instead, the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical? (c) If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at Figure P7.74 which the ball can move?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Centripetal Force

A particle, body, object or certain mass while travelling in a rotational path or semicircular path, experiences an inertia force. If that inertia force pulls the towards the inward of the imaginary circle in the rotation, it is termed as centripetal force.

Question

**Problem Context:**

A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane, with the cord making a 30° angle with the vertical.

**Objectives:**

(a) Determine the ball’s speed.

(b) If, instead, the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical?

(c) If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at which the ball can move?

**Diagram Overview:**

- The diagram shows a hand holding a cord at an angle $\theta$ to the vertical.
- The ball is revolving in a circular path, indicated by a dashed circle.

**Considerations:**

- The cord length is 1.5 m.
- The angle with the vertical is initially 30°.
- Use these parameters to solve the questions and understand the dynamics of circular motion.

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