A "swing" ride at a carnival consists of chairs that are swung in a circle by 17.4 m cables attached to a vertical rotating pole, as the drawing shows. Suppose the total mass of a chair and its occupant is 291 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.
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To determine the tension in the cable attached to the chair, we need to consider the forces acting on the chair in equilibrium. The tension in the cable is the force that counteracts the force of gravity on the chair and its occupant.
(a) Tension in the cable attached to the chair:
1. Begin by drawing a free-body diagram representing the forces acting on the chair. On this diagram, label the forces as follows:
- Tension in the cable (T) - upwards
- Force of gravity (mg) - downwards
2. Write the equations for the forces acting on the chair in the vertical direction:
ΣFy = T - mg = 0
3. Since the chair is in equilibrium (not accelerating vertically), the sum of the forces in the vertical direction must equal zero. Therefore:
T = mg
4. Substitute the known values into the equation:
T = (291 kg) * (9.8 m/s²) (acceleration due to gravity ≈ 9.8 m/s²)
T ≈ 2851.8 N
Thus, the tension in the cable attached to the chair is approximately 2851.8 N.
(b) Speed of the chair:
To find the speed of the chair, we need to consider the centripetal force acting on the chair. The centripetal force is provided by the tension in the cable.
1. The centripetal force acting on the chair is given by:
Fc = mv² / r
Where:
- Fc is the centripetal force
- m is the mass of the chair and occupant (291 kg)
- v is the velocity (speed) of the chair (what we need to find)
- r is the radius of the circular path (17.4 m)
2. Since the tension in the cable provides the centripetal force, we can set the equations for the centripetal and tension forces as equal to each other:
T = Fc
3. Substituting the equations for centripetal force and tension:
T = mv² / r
mv² / r = mg
4. Rearrange the equation to solve for v:
v² = rg
v = √( rg )
5. Substitute the known values and solve for v:
v ≈ √( (17.4 m) * (9.8 m/s²) )
v ≈ √( 170.52 m²/s² )
v ≈ 13.05 m/s
Therefore, the speed of the chair is approximately 13.05 m/s.