A 500 kg piano is being lowered into position by a crane...

A 500 kg piano is being lowered into position by a crane while two people steady it with ropes pulling to the sides. Bob's rope pulls to the left, $15^{\circ}$ below horizontal, with $500 \mathrm{N}$ of tension. Ellen's rope pulls toward the right, $25^{\circ}$ below horizontal. a. What tension must Ellen maintain in her rope to keep the piano descending at a steady speed? b. What is the tension in the main cable supporting the piano?

A 500 kg piano is being lowered into position by a crane while two people steady it with ropes pulling to the sides. Bob's rope pulls to the left, $15^{\circ}$ below horizontal, with $500 \mathrm{N}$ of tension. Ellen's rope pulls toward the right, $25^{\circ}$ below horizontal.
a. What tension must Ellen maintain in her rope to keep the piano descending at a steady speed?
b. What is the tension in the main cable supporting the piano?

Key Concepts

Force Decomposition into Components

Force decomposition is the process of breaking a force vector into its horizontal and vertical components using trigonometric functions such as sine and cosine. This technique is essential when dealing with forces applied at angles and allows for the independent application of equilibrium conditions along each axis.

Tension in Ropes and Cables

Tension is the pulling force transmitted along a rope or cable when forces are applied at its ends. Understanding how to calculate tension using vector components and equilibrium conditions is crucial in engineering and physics problems involving suspended or supported loads.

Application of Newton's First Law

Newton's first law states that an object will remain at rest or move at a constant velocity unless acted upon by an unbalanced force. This law underlies the concept of static equilibrium, where the sum of forces in each direction must be zero for a system moving steadily.

Static Equilibrium

Static equilibrium refers to a state where all forces and moments acting on an object cancel, resulting in no net acceleration. This principle is used when analyzing systems that remain at rest or move with constant velocity, ensuring that the forces in both horizontal and vertical directions balance.

Free-Body Diagrams

Free-body diagrams are simplified representations of objects isolated from their environment, showing all the forces acting on them. These diagrams are key for visually organizing and setting up the equations required to solve for unknown variables in statics and dynamics.

Transcript

00:01 In this question, we are asked about a 500 -kilogram piano being lowered into position by a crane, and two people are studying it with ropes pulling to the sides.

00:13 One rope is pulling to the left, 15 degrees below the horizontal with 500 newtons of tension, and the second rope is pulled towards the right, 25 degrees below the horizontal.

00:26 We're asked to find the tension in the rope and ellen's rope pulling towards the right and as well as the tension in the main cable from the crane supporting the piano.

00:37 So i think the best way to start a question like this is just with a free body diagram.

00:45 So we have the tension in the main rope directed upwards.

00:52 And i'm going to call that tc for t crane.

00:56 And then we have the gravitational force directed down as always.

01:00 And then we have the rope pulling to the left.

01:08 Bob pulls this rope, so i'm going to call this tb, and it makes an angle of 25 degrees.

01:18 And then we have, oh, sorry, not 25, this one is 15.

01:25 And then we have the one that is directed to the left or to the right, and it's directed 25 degrees below the horizontal.

01:36 And i'm going to call this te for ellen, since ellen is the one holding this rope.

01:42 So now that we've got a nice free body diagram, we can use our force balance equations to solve for the forces that were asked for in this problem.

01:55 So in particular, we know that the crane is being lowered steadily.

02:04 And we were asked to find the force from ellen in order to keep the piano descending at a steady speed.

02:13 So what that's telling us is that the speed here is constant, and so we can assume the acceleration is zero, and that the net force is also zero.

02:27 So i'm going to do two force balance equations.

02:30 The first one will be for the x direction, and that's going to be equal to zero.

02:36 Right f net equals m a is zero so in the x direction what forces do we have we have the x components of bob that's pointing to the left so i'm going to put a minus sign in front of it and then we have the x component from ellen that's pointing in the to the right so i'm going to put a positive leave that as a positive and so that tells us that the tension that ellen pulls with in the x direction has to be equal to the tension that bob pulls with in the x direction.

03:14 So we can solve these, both of these, for its components.

03:20 So t .e.

03:21 X is going to be t .e.

03:24 Kos 25.

03:27 And t .b .x will be t .b .k .c...

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