A piñata of mass M=8.00 kg is attached to a rope of...

A piñata of mass $M=8.00 \mathrm{~kg}$ is attached to a rope of negligible mass that is strung between the tops of two vertical poles. The horizontal distance between the poles is $D=2.00 \mathrm{~m},$ and the top of the right pole is a vertical distance $h=0.500 \mathrm{~m}$ higher than the top of the left pole. The piñata is attached to the rope at a horizontal position halfway between the two poles and at a vertical distance $s=1.00 \mathrm{~m}$ below the top of the left pole. Find the tension in each part of the rope due to the weight of the piñata.

Key Concepts

Static Equilibrium

Static equilibrium is the condition in which a system experiences no net force or torque, meaning that all forces, including gravitational, tension, and any reaction forces, balance out perfectly. This principle is fundamental in analyzing structures and objects at rest, ensuring that every applied force has an equal and opposing force to maintain stability.

Free-Body Diagrams

A free-body diagram is a simplified illustration that isolates an object and represents all the forces acting upon it. This tool is crucial for visualizing the interactions in a system, breaking down forces into their components, and applying the principles of static equilibrium effectively.

Vector Decomposition

Vector decomposition involves breaking a force into its horizontal and vertical components. This process is essential for analyzing forces acting at angles, as it allows the application of equilibrium conditions separately in each direction, facilitating the accurate calculation of forces like tension.

Tension in Ropes

Tension in ropes refers to the force transmitted along a rope when it is pulled tight by forces acting from opposite ends. This concept is pivotal in problems involving suspended loads, where different segments of the rope may experience different magnitudes of tension due to variations in angles and points of application of forces.

Geometric Considerations

Geometric considerations include analyzing the spatial configuration of a system, such as the distances, angles, and relative positions of attachment points. These considerations are critical in resolving forces into their components and understanding how the physical layout affects the balance of forces in the system.

Transcript

00:01 This problem we have given a panata of mass 8 kg is 8 kg is attached to a rope of negligible mass that is strung between the tops of two vertical poles.

00:15 This is left pole and this is right pole the distance between these two poles is 2 meter and this right pole is given it is 0 .5 meter higher than the top of the left pole so this is 0 .5 meter and it is given that the penanta is attached to the rope at a horizontal position halfway between the two poles that means on this line and a vertical distance one meter below the top of the left pole so so here is suppose finanta this distance is 1 meter and this distance is also 1 meter like this this rope is connected this distance is 1 .5 meter also sorry 1 meter because it is half way between the two poles so it is 1 meter now we can find this angle theta theta is equal to 45 degree right we have to find the tension in each part of the rope due to weight of the finata right so we know that suppose tension in this ring is t1 and this is t2 and suppose this angle is alpha okay this distance is 1 meter so this total distance is 1 .5 meter so so we can find this alpha angle also.

02:26 We know that 10 alpha will be equal to 1 .5 divided by 1.

02:33 So alpha will be equal to 10 inverse 1 .5.

02:38 So this is equal to 10 inverse 1 .5 which is equal to 56 .3 degree.

02:54 This alpha is 56 .3 degree.

02:57 We know there this penanta is in equilibrium position.

03:01 So the net force in horizontal direction should be equal to 0.

03:06 Net force in horizontal direction should be equal to 0.

03:10 So the horizontal component of t1, which is t1 cos theta, should be equal to t2, cos alpha...