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8. Centripetal Forces & Gravitation
Centripetal Forces
6:54 minutes
Problem 5b
Textbook Question
Textbook QuestionThe 'Giant Swing' at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable 5.00 m long, and the upper end of the cable is fastened to the arm at a point 3.00 m from the central shaft (Fig. E5.50). (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of 30.0° with the vertical.
Verified step by step guidance
1
Step 1: Identify the forces acting on the seat. The forces are the tension in the cable (T) and the gravitational force (mg).
Step 2: Resolve the tension into its vertical (Tcos(θ)) and horizontal (Tsin(θ)) components, where θ is the angle the cable makes with the vertical.
Step 3: Use the vertical component of the tension to balance the gravitational force: Tcos(θ) = mg.
Step 4: Use the horizontal component of the tension to provide the centripetal force: Tsin(θ) = m(v^2/r), where v is the tangential speed and r is the radius of the circular path.
Step 5: Combine the equations from steps 3 and 4 to solve for the tangential speed (v), and then use the relationship between the tangential speed, radius, and period (T) of the circular motion: v = 2πr/T.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Centripetal Force
Centripetal force is the net force that acts on an object moving in a circular path, directed towards the center of the circle. In the context of the Giant Swing, this force is necessary to keep the seat moving in a circular motion as it swings around the central shaft. It is provided by the tension in the cable and the gravitational force acting on the seat.
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Pendulum Motion
Pendulum motion refers to the oscillatory movement of a mass attached to a fixed point by a cable or rod. In this scenario, the seat of the Giant Swing acts like a pendulum, swinging back and forth as it rotates around the central shaft. The angle of the cable with the vertical affects the period of the swing, which is the time taken for one complete revolution.
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Trigonometric Relationships
Trigonometric relationships are mathematical functions that relate the angles of a triangle to the lengths of its sides. In this problem, the angle of 30° with the vertical can be used to determine the horizontal and vertical components of the forces acting on the seat. These relationships are crucial for calculating the radius of the circular path and the effective forces involved in the swing's motion.
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