(II) A 110-kg horizontal beam is supported at each end. A 320-kg piano rests a quarter of the wa

step 1 All right, chapter 9, problem 9. We have a piano sitting on two beams, and we basically want to

(II) A 110-kg horizontal beam is supported at each end. A...

Key Concepts

Moments and Lever Arms

Moments, or torques, are produced when a force is applied at a distance from a pivot point. The lever arm is the perpendicular distance from the pivot to the line of action of the force. In static equilibrium problems, the sum of the moments about any point must also be zero, allowing for the determination of unknown forces by balancing the clockwise and counterclockwise moments.

Center of Mass (of the Beam)

The center of mass is the point at which the mass of the beam can be considered to be concentrated. For a uniform beam, this is at its geometric center. Recognizing the center of mass is important as it helps determine the moment due to the beam’s own weight, which affects the equilibrium of the system.

Static Equilibrium

Static equilibrium is the state where an object is at rest and the sum of all forces and moments (torques) acting on it is zero. This principle ensures that the system experiences no net motion or rotation, which is critical when analyzing structures like beams supported at their ends under various loads.

Sum of Forces (Vertical Forces)

In problems of static equilibrium, the sum of the vertical forces must be zero. This means that the total upward support forces provided by the supports must exactly counterbalance the downward gravitational forces acting on the beam and any additional loads, ensuring that there is no net vertical motion.

Reaction Forces

Reaction forces are the forces exerted by supports or connections when an object is in equilibrium. In a beam supported at both ends, reaction forces are the upward forces that balance the downward gravitational loads. Calculating these forces involves setting up equilibrium equations for both the vertical forces and the moments around a chosen pivot point.

Transcript

00:01 All right, chapter 9, problem 9.

00:06 We have a piano sitting on two beams, and we basically want to find the forces on the forces.

00:12 Well, let's read the question here.

00:14 Yes, what is the vertical force on each of the supports? okay, so we want to find the vertical force on each.

00:19 We want to find both of these forces.

00:23 And so if we draw a picture here, two beams, on the left and the right, so focusing on above the line, above this line here, above this line here.

00:34 Two beams on the left and right, a piano, a quarter from the left, and another dot here at the center of the beam for the center of mass.

00:43 We can basically draw a free body diagram that allows us to calculate torques, which is going to become useful later on.

00:53 So here's our free body diagram below.

00:56 The left support is going to push upwards on the beam.

01:00 The piano is going to push downwards due to gravity, right, with a big body.

01:04 M so this is a big m for its mass 320 kilograms.

01:09 The center of gravity, the beam is going to, is massive itself and is going to feel gravity, so it has a little m, g, that's 110, kilograms for the mass.

01:19 And then the right beam also is going to have an upward force.

01:22 And then the two lengths we're going to be concerned with are the length, complete length of the beam, just l, and the quarter length from the distance from the left beam to the piano.

01:36 Eventually we want to use newton's second law here.

01:41 Since we're in equilibrium, since the piano's just sitting on it not moving, no acceleration, then we're allowed to say that the sum of the forces is equal to zero, but also the sum of the torques is equal to zero.

01:51 Because they both deal with acceleration.

01:54 So if we're talking about a pivot, that means the angular acceleration is zero.

01:57 There's no rotation.

01:59 And if we're talking about just a linear force, a linear movement, that means that the linear acceleration is zero so this is just linear and this is rotational right but with that we can solve and find a lot of things so here we go if we take the left beam as our pivot and counterclockwise as our positive direction then the following is true zero equals the sum of the torques and this is also equal to the the right force the force from the right beam multiplied by the full distance minus m g times l over 2 minus big m g times l over 4 so where do these come from well this is just torque right torque is multiple torque is r cross f that's the vector form to make it a little bit simpler it's just forces acting at a distance so since we chose the left part to be our pivot any force acting from that distance is going to cause a torque so the two weights are going to cause a downward torque so they're going to be negative and then the right beam force is going to cause upward torque, it's going to be positive...

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