Question

An overhead crane is suspended from a ceiling by two chains. One chain is 4.6 m long and forms an angle of 60° with the ceiling. The other chain is 6.4 m long. What angle does the larger chain make with the ceiling?

(A) 42.8°
(B) 47.2°
(C) 60.0°
(D) 68.2°

Answer 1

Despite the difference in chain lengths, the longer chain also forms a 60° angle with the ceiling because the horizontal components of the tension must be the same for both. Therefore, the answer is option (C).

Step-by-step explanation:

The student asked what angle the larger chain (6.4 m long) makes with the ceiling when an overhead crane is suspended from a ceiling by two chains, where one chain is 4.6 m long and forms an angle of 60° with the ceiling. Since the chains are supporting the same crane, the horizontal components of the tension in the chains must be equal because they counterbalance each other.

This means we can set up the following relationship using trigonometric functions:

• The horizontal component of the tension in the shorter chain (Chain 1 at 4.6 m) is T1×cos(60°).
• The horizontal component of the tension in the longer chain (Chain 2 at 6.4 m) is T2×cos(θ), where θ is the angle we are trying to find.

Equating these two horizontal components gives us:

T1×cos(60°) = T2×cos(θ)

Since T1 and T2 are the tensions in the chains, and the crane is stationary, we can assume that the tensions are equal (T1 = T2) because they both support the same crane's weight. Thus, we can simply solve for θ as:

cos(60°) = cos(θ)

cos(θ) = 0.5, hence θ = cos⁻¹(0.5) = 60°.

This means that, surprisingly, the longer chain also makes a 60° angle with the ceiling, which is option (C).

Answer 2

The angle the larger chain makes with the ceiling is 60°, as the chains and ceiling form a triangle and the sum of angles must be 180°.

Step-by-step explanation:

To determine the angle the longer chain makes with the ceiling, we need to recognize that the chains and the ceiling form a triangle. Since one angle is already given as 60°, we know that the angle between the 4.6 m chain and the ceiling is 120° because the total sum of angles in a straight line is 180°. The sum of angles in any triangle must be 180°, so the sum of the angle made by the longer chain with the ceiling and the given 120° angle must be 60° (180° - 120°). Therefore, the angle the larger chain makes with the ceiling is 60°, which corresponds to option (C).