Two ropes pull on a ring. One exerts a 63 N force at 30.0°, the other a 63 N force at 60.0°
(a) What is the net force on the ring?
Magnitude
____ N
Direction
____ °
(b) What are the magnitude and direction of the force that would cause the ring to be in equilibrium?
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To find the net force on the ring, we need to analyze the two forces acting on it.
(a) Net Force:
1. Start by resolving each force into horizontal and vertical components using trigonometry.
- For the first force: 63 N at 30.0°, the horizontal component (Fx₁) will be F * cos(θ) and the vertical component (Fy₁) will be F * sin(θ).
- For the second force: 63 N at 60.0°, the horizontal component (Fx₂) will be F * cos(θ) and the vertical component (Fy₂) will be F * sin(θ).
2. Add the horizontal components together to find the horizontal net force (Fnetx).
Fnetx = Fx₁ + Fx₂.
3. Add the vertical components together to find the vertical net force (Fnety).
Fnety = Fy₁ + Fy₂.
4. Use the Pythagorean theorem to find the magnitude of the net force (Fnet).
Fnet = √(Fnetx² + Fnety²).
5. Use trigonometry to find the direction of the net force (θnet).
θnet = arctan(Fnety / Fnetx).
Now let's calculate the values:
For the first force (63 N at 30.0°):
Fx₁ = 63 N * cos(30.0°) ≈ 54.45 N
Fy₁ = 63 N * sin(30.0°) ≈ 31.50 N
For the second force (63 N at 60.0°):
Fx₂ = 63 N * cos(60.0°) ≈ 31.50 N
Fy₂ = 63 N * sin(60.0°) ≈ 54.45 N
Fnetx = Fx₁ + Fx₂ ≈ 54.45 N + 31.50 N ≈ 85.95 N
Fnety = Fy₁ + Fy₂ ≈ 31.50 N + 54.45 N ≈ 85.95 N
Fnet = √(Fnetx² + Fnety²) ≈ √(85.95 N)² + (85.95 N)²) ≈ √14655 N² ≈ 121 N (rounded to 3 significant figures)
θnet = arctan(Fnety / Fnetx) ≈ arctan(85.95 N / 85.95 N) ≈ 45.0°
Therefore, the net force on the ring is approximately 121 N in a direction of 45.0°.
(b) To find the magnitude and direction of the force that would cause the ring to be in equilibrium, we need a force that exactly cancels out the net force.
Magnitude:
The magnitude of the force needed to achieve equilibrium is equal to the magnitude of the net force, which is approximately 121 N.
Direction:
The direction of the force needed to achieve equilibrium is the opposite of the direction of the net force. Since the net force is at an angle of 45.0°, the force needed for equilibrium should be at an angle of 45.0° + 180° = 225.0°.
Therefore, the magnitude of the force for equilibrium is approximately 121 N, and the direction is approximately 225.0°.