The tightrope walker gets tired and decides to stop for a rest. During this rest period, she is in translational equilibrium. She stops at middle of the rope and finds that both sides of the rope make an angle of è = 10.00° with the horizontal. If the mass of the tightrope walker is 60 kg, what is the tension in the rope?
2Tsinα=mg
T=mg/2sinα
To find the tension in the rope, we can analyze the forces acting on the tightrope walker when she is at rest.
1. Start by drawing a diagram of the situation. The tightrope walker is at the middle of the rope, which makes an angle of 10.00° with the horizontal on both sides. Label this angle as θ.
Tension (T)
<----θ---->
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| Tightrope Walker |
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<----θ---->
Tension (T)
2. Since there is no acceleration in the vertical direction (translational equilibrium), the vertical component of the tension on each side of the rope must balance the weight of the tightrope walker.
3. Calculate the vertical component of the tension. This can be done using the trigonometric function sine (sin).
sin(θ) = Opposite / Hypotenuse
The vertical component of the tension is the opposite side, and the hypotenuse is the tension itself. Therefore,
sin(θ) = (Vertical Component of Tension) / Tension
Rearrange the equation to solve for the vertical component of the tension:
(Vertical Component of Tension) = Tension * sin(θ)
4. Calculate the weight of the tightrope walker:
Weight (W) = mass * acceleration due to gravity
W = 60 kg * 9.8 m/s^2
5. The weight of the tightrope walker is balanced by the vertical components of the tension on both sides of the rope. Since the weight is acting downward, the vertical components of the tension must be acting upward with the same magnitude.
Therefore,
(Vertical Component of Tension) = Weight / 2
(Vertical Component of Tension) = (60 kg * 9.8 m/s^2) / 2
6. Plug in the values and solve for the vertical component of the tension:
(Vertical Component of Tension) = (60 kg * 9.8 m/s^2) / 2
(Vertical Component of Tension) = 294 N
7. Finally, to find the actual tension in the rope, we can use the trigonometric function cosine (cos).
cos(θ) = Adjacent / Hypotenuse
The horizontal component of the tension is the adjacent side, and the hypotenuse is the tension itself.
Rearrange the equation to solve for the tension:
Tension = (Horizontal Component of Tension) / cos(θ)
Since the tension is the same on both sides of the rope:
Tension = 2 * (Horizontal Component of Tension) / cos(θ)
Tension = 2 * (294 N) / cos(10.00°)
8. Plug in the values and solve for the tension in the rope:
Tension = 2 * (294 N) / cos(10.00°)
Tension ≈ 601 N
Therefore, the tension in the rope is approximately 601 N.
To find the tension in the rope, we can use the concept of forces in equilibrium. When the tightrope walker is at rest in the middle of the rope, the forces acting on her must be balanced for her to be in translational equilibrium.
Let's analyze the forces acting on the tightrope walker:
1. Weight (W): This is the force due to gravity acting vertically downwards. The magnitude of the weight can be calculated using the formula W = m * g, where m is the mass of the tightrope walker and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Tension in the rope (T): This is the force exerted by the rope on the tightrope walker. It acts along the direction of the rope and keeps the walker from falling.
Now, let's resolve the weight into its components:
- The component of weight parallel to the rope (W_parallel) is balanced by the tension in the rope (T).
- The component of weight perpendicular to the rope (W_perpendicular) will balance out due to the normal forces from the rope and the ground, as the tightrope walker is in equilibrium.
Since both sides of the rope make an angle of 10.00° with the horizontal, the angle between the weight (W) and the rope will be 90° - 10° = 80°.
Now, let's calculate the component of weight parallel to the rope:
W_parallel = W * sin(80°)
Substituting the values:
W_parallel = (60 kg) * (9.8 m/s^2) * sin(80°)
Now, we can equate the tension in the rope (T) to the component of weight parallel to the rope (W_parallel):
T = W_parallel
Therefore, the tension in the rope is:
T = (60 kg) * (9.8 m/s^2) * sin(80°)
Calculating this value will give us the tension in the rope.