-4.45 A pinata of mass M=8.0 kg is attached to a rope of negligible mass that is strung between

-4.45 A pinata of mass M=8.0 kg is attached to a rope of...

-4.45 A pinata of mass $M=8.0 \mathrm{~kg}$ 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 $D=2.0 \mathrm{~m},$ and the top of the right pole is a vertical distance $h=0.50 \mathrm{~m}$ higher than the top of the left pole. The pinata is attached to the rope at a horizontal position halfway between the two poles and at a vertical distance $s=1.0 \mathrm{~m}$ below the top of the left pole. Find the tension in each part of the rope due to the weight of the pinata.

Key Concepts

Static Equilibrium

This is the principle that an object at rest has all forces balanced, meaning the net force acting on it is zero. In problems like these, applying static equilibrium conditions allows us to set up equations where the sum of vertical and horizontal force components equals zero, serving as the foundation for determining unknown forces such as tensions.

Free-Body Diagrams

A free-body diagram is a visual tool used to isolate a body and represent all the forces acting upon it. It helps identify all the applied forces, such as weight and tension in ropes, and their directions, which is crucial for setting up correct equilibrium equations in mechanics problems.

Resolving Forces into Components

Many physics problems require breaking down forces into their horizontal and vertical components, especially when forces act at an angle. This process simplifies complex problems by allowing separate equilibrium equations for each direction, which can then be solved for unknown quantities like the tension in different segments of a rope.

Trigonometry in Physics

Trigonometry is essential for relating the geometry of a problem (angles and distances) to the physics (force components). It enables determination of the vertical and horizontal contributions of forces acting along inclined lines, providing the necessary relationships to solve for quantities like rope tension.

Newton's Laws of Motion

Newton's laws, particularly the first law regarding equilibrium, govern the behavior of forces in static situations. They state that the sum of all forces acting on a stationary object must be zero, offering a fundamental basis for constructing equations that describe the balance of forces in systems such as a rope supporting a mass.

Transcript

00:01 So here we'll be able to use trigonometry in order to figure out these angle measures.

00:06 We can say that theta sub 1 would be equaling to arc tan of length s divided by then d over 2.

00:18 We have then this would be equal to arc tan of 1 .0 meters divided by 1 .0 meters.

00:28 This is of course 1, arc tan of 1, 45 degrees.

00:32 We have theta sub 2.

00:40 This would be equal to then arc tan, and this would be s plus h.

00:47 This would then be divided by d over 2.

00:50 And this would give us then arc tan of 1 .5 meters divided by 1 .0 meters, giving us then 56 .31 degrees.

01:06 Now we can apply newton's second law in the xmr directions, sum of forces, the x direction we have translational equilibrium in the x direction so we can say that this would be equal to zero t sub 1 cosine of theta sub 1 minus t sub 2 cosine of theta sub 2 this would be equal to then 0 and solving and isolating t sub 1 t sub 1 would then be equal to t sub 2 multiplied by cosine of a theta sub 2 divided by cosine of a theta sub 1 the sum of forces in the y direction would equal the mass times the acceleration of the y direction again we have translational equilibrium in the in the in the direction so this will be equal to 0 and we have t sub 1 sign of theta sub 1 plus t sub 2 sine of theta sub 2 minus m g equaling then zero.

02:22 We can rearrange quickly and say that the two tension, the two y components of the two tension forces is simply going to be equal to the gravitational force.

02:37 And so we're going to plug one in, t sub one in, from the x equation into the y equation, and say that then t sub 2, cosine of theta sub 2, divided by cosine of theta sub 1...

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