The figure shows a stationary arrangement of two crayon boxes and three cords. Box A has a mass of 12.6 kg and is on a ramp at angle θ = 28.9o; box B has a mass of 6.06 kg and hangs on a cord. The cord connected to box A is parallel to the ramp, which is frictionless. (a) What is the tension in the upper cord, and (b) what angle does that cord make with the horizontal?
To find the tension in the upper cord (T1), we need to consider the forces acting on box A.
1. Determine the forces acting on box A:
- Weight of box A (W_A): W_A = m_A * g, where m_A is the mass of box A and g is the acceleration due to gravity.
- Tension in the upper cord (T1): This is the force being applied to keep the box from sliding down the ramp.
- Normal force (N): This force is perpendicular to the ramp and balances the component of the weight of the box that is perpendicular to the ramp.
- Friction force (f): Since the ramp is frictionless, this force is negligible.
2. Resolve the weight of box A:
The weight of box A can be resolved into two components: one parallel to the ramp and one perpendicular to the ramp.
- Weight parallel to the ramp (W_parallel): W_parallel = W_A * sin(θ)
- Weight perpendicular to the ramp (W_perpendicular): W_perpendicular = W_A * cos(θ)
3. Sum the forces in the vertical direction:
- ΣF_Y = T1 - N - W_perpendicular = 0
Since the box is stationary in the vertical direction, the sum of the forces in that direction is zero. Therefore, we can solve for N:
N = T1 - W_perpendicular
4. Sum the forces in the horizontal direction:
- ΣF_X = T1 - W_parallel = 0
Again, since the box is stationary in the horizontal direction, the sum of the forces in that direction is zero. Therefore, we can solve for T1:
T1 = W_parallel
5. Calculate the values:
- Calculate W_parallel: W_parallel = W_A * sin(θ)
- Calculate W_perpendicular: W_perpendicular = W_A * cos(θ)
- Substitute the values into the equations above to find T1.
To find the angle that the cord makes with the horizontal, we need to consider the forces acting on box B and use trigonometry.
6. Determine the forces acting on box B:
- Weight of box B (W_B): W_B = m_B * g, where m_B is the mass of box B and g is the acceleration due to gravity.
- Tension in the lower cord (T2): This is the force being applied upwards by the hanging box.
7. Resolve the weight of box B:
The weight of box B acts vertically downwards.
8. Sum the forces in the vertical direction:
- ΣF_Y = T2 - W_B = 0
Since the box is stationary in the vertical direction, the sum of the forces in that direction is zero. Therefore, we can solve for T2:
T2 = W_B
9. Find the angle that the cord makes with the horizontal:
- Use trigonometry to determine the angle. The angle can be found using the inverse tangent function:
Angle = atan(W_parallel / W_perpendicular)
10. Substitute the values into the equation above to find the angle.