Question

a crate hangs from a rope that is attached t a metal ring. the metal ring is suspended by a scond rope that is attached overheat at two points, as shonw. what is the angle if the tension in rope 1 is 1.74 times the tension in rope 2?

Answer 1

The angle between rope 1 and the vertical is approximately 29.54 degrees.

How to find angle?

The tension in rope 1 (
`\( T_1 \)`) is 1.74 times the tension in rope 2 (
`\( T_2 \)`). This gives us the relationship:

`\[ T_1 = 1.74 * T_2 \]`

Since the metal ring is in static equilibrium, the vertical components of the tension in rope 2 must balance the tension in rope 1.

Let θ be the angle each part of rope 2 makes with the vertical, the vertical force balance:

`\[ 2T_2 \cos(\theta) = T_1 \]`

Substituting
`\( T_1 \)` from the relationship:

`\[ 2T_2 \cos(\theta) = 1.74 * T_2 \]`

The
`\( T_2 \)` terms cancel out, because they are non-zero, leaving an equation involving just θ:

`\[ 2 \cos(\theta) = 1.74 \]`

Divide both sides by 2 to isolate cos(θ):

`\[ \cos(\theta) = (1.74)/(2) \]`

`\[ \cos(\theta) = 0.87 \]`

Solve for θ using the inverse cosine function:

`\[ \theta = \cos^(-1)(0.87) \]`

Two possible angles due to the symmetry of the cosine function around 0 degrees (or 360 degrees). These are:

`\[ \theta \approx 29.54^\circ \]`

or

`\[ \theta \approx 360^\circ - 29.54^\circ \]`

`\[ \theta \approx 330.46^\circ \]`

Since it is the acute angle between the rope and the vertical that is unknown, take the acute angle, which is θ ≈ 29.54°

Therefore, the angle between rope 1 and the vertical is approximately 29.54°