Questions: 9) A 65-kg gymnast hangs from a pair of rings attached to ropes, as shown. In figure (i), the ropes are vertical. In figure (ii), the ropes make an angle of 55° with the ceiling. First for (i), and then repeat for (ii): a) Draw a FBD of the gymnast. b) Determine the components of each force vector from your FBD (in symbols). c) Determine the magnitude of the tension in the ropes. (i) (ii) (Solve in symbols, and then plug in numbers at the end.)

9) A 65-kg gymnast hangs from a pair of rings attached to ropes, as shown. In figure (i), the ropes are vertical. In figure (ii), the ropes make an angle of 55° with the ceiling. First for (i), and then repeat for (ii):
a) Draw a FBD of the gymnast.
b) Determine the components of each force vector from your FBD (in symbols).
c) Determine the magnitude of the tension in the ropes.
(i) (ii) (Solve in symbols, and then plug in numbers at the end.)

Transcript text: 9) A $65-\mathrm{kg}$ gymnast hangs from a pair of rings attached to ropes, as shown. In figure (i), the ropes are vertical. In figure (ii), the ropes make an angle of $55^{\circ}$ with the ceiling. First for (i), and then repeat for (ii): a) Draw a FBD of the gymnast. b) Determine the components of each force vector from your FBD (in symbols). . c) Determine the magnitude of the tension in the ropes. (i) (ii) (Solve in symbols, and then plug in numbers at the end.)

Solution

Solution Steps

Step 1: Draw a Free Body Diagram (FBD) of the Gymnast

For both scenarios (i) and (ii), draw the gymnast hanging from the ropes.
In scenario (i), the ropes are vertical, so the tension forces are directly upward.
In scenario (ii), the ropes make an angle of 55° with the ceiling, so the tension forces are at an angle.

Step 2: Determine the Components of Each Force Vector from Your FBD (in symbols)

For scenario (i):
- The tension forces $ T_1 $ and $ T_2 $ are vertical.
- The weight $ W $ of the gymnast acts downward.
- $ T_1 + T_2 = W $.
For scenario (ii):
- The tension forces $ T_1 $ and $ T_2 $ are at an angle of 55° with the ceiling.
- The vertical components of the tension forces are $ T_1 \cos(55^\circ) $ and $ T_2 \cos(55^\circ) $.
- The horizontal components of the tension forces are $ T_1 \sin(55^\circ) $ and $ T_2 \sin(55^\circ) $.
- The sum of the vertical components must equal the weight of the gymnast: $ T_1 \cos(55^\circ) + T_2 \cos(55^\circ) = W $.
- The horizontal components must cancel each other out: $ T_1 \sin(55^\circ) = T_2 \sin(55^\circ) $.

Step 3: Determine the Magnitude of the Tension in the Ropes

For scenario (i):
- Since the ropes are vertical and the gymnast is in equilibrium, the tension in each rope is half the weight of the gymnast.
- $ T_1 = T_2 = \frac{W}{2} = \frac{mg}{2} $.
- Given $ m = 65 $ kg and $ g = 9.8 $ m/s², $ T_1 = T_2 = \frac{65 \times 9.8}{2} = 318.5 $ N.
For scenario (ii):
- Using the vertical component equation: $ 2T \cos(55^\circ) = W $.
- Solving for $ T $: $ T = \frac{W}{2 \cos(55^\circ)} = \frac{mg}{2 \cos(55^\circ)} $.
- Given $ m = 65 $ kg and $ g = 9.8 $ m/s², $ T = \frac{65 \times 9.8}{2 \cos(55^\circ)} $.
- Calculate $ \cos(55^\circ) \approx 0.5736 $.
- $ T = \frac{65 \times 9.8}{2 \times 0.5736} \approx 556.4 $ N.