Final answer:
When an object hangs from two ropes at unequal angles, the tension in each rope will differ to counteract the gravitational force, leading to a state of equilibrium. Free-body diagrams and trigonometry are essential tools used to calculate the individual tensions in the ropes.
Step-by-step explanation:
When an object of mass M hangs from two ropes at unequal angles, the tension in each rope will differ to balance the gravitational force acting downward on the mass. This situation can be analyzed using a free-body diagram. In such a case, the force of tension acts upward along the ropes, while the gravitational force (weight) acts downward. The tension forces in the rope are at different angles, and their vector sum needs to equal the gravitational force for the system to be in equilibrium. Calculating the tension in each rope requires the use of trigonometry, breaking the tensions into their horizontal and vertical components and applying Newton's second law.
For example, in a scenario where the mass hangs stationary from ropes, the net force must be zero. This condition requires that the sum of the vertical components of the tensions in both ropes equals the weight of the mass (Mg), and the sum of the horizontal components of the tensions cancel each other out. Through mathematical equations, one can solve for the tension in each rope when given the angles at which the ropes are hanging and the mass M.