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Chapter 11

Chapter 11. Equilibrium and Elasticity. Goals for Chapter 11. To study the conditions for equilibrium. Homework. Read 404 – 414 1, 5, 11, 17, 21. Introduction.

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Chapter 11

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  1. Chapter 11 Equilibrium and Elasticity

  2. Goals for Chapter 11 • To study the conditions for equilibrium

  3. Homework • Read 404 – 414 • 1, 5, 11, 17, 21

  4. Introduction • If we find all forces and torques in equilibrium, we can understand situations we see every day … ladders firmly against a wall, bridges, even hanging pictures.

  5. Conditions for equilibrium • The sum of all forces present in the x, y, and z directions are each distinctly equal to zero. • The sum of all torques for any given point are equal to zero.

  6. The center of gravity (COG) • Calculations begin with the model of all forces and torques acting on the center of gravity for an object. • For a straightforward object like a solid sphere of uniform density, the center of gravity is … the center of the sphere.

  7. A “teeter-totter” balancing torques • A uniform wooden plank of length L = 6.0 m and mass M = 90 kg rests on top of two sawhorses separated by D = 1.5 m, located equal distances from the center of the plank. Your cousin Throckmorton tries to stand on the right-hand end of the plank. If the plank is to remain at rest, how massive can Throckmorton be?

  8. Problem Solving Strategy • IDENTIFYthe relevant concepts: The first and second conditions for equilibrium are useful whenever there is a rigid body that is not rotating and not accelerating in space. • SET UPthe problem using the following steps:   • 1. Draw a sketch of the physical situation, including dimensions, and select the body in equilibrium to be analyzed. • 2. Draw a free-body diagram showing the forces acting on the selected body and no others. Do not include forces exerted by this body on other bodies. Be careful to show correctly the point at which each force acts; this is crucial for correct torque calculations. You can’t represent a rigid body as a point. • 3. Choose coordinate axes and specify a positive direction of rotation for torques. Represent forces in terms of their components with respect to the axes you have chosen; when you do this, cross out the original force so that you don’t include it twice.

  9. Problem Solving Strategy • SET UPcontinued:   • In choosing a point about which to compute torques, note that if a force has a line of action that goes through a particular point, the torque of the force with respect to that point is zero. You can often eliminate unknown forces or components from the torque equation by a clever choice of point for your calculation. The body doesn’t actually have to be pivoted about an axis through the chosen point. • EXECUTEthe solution as follows:   • 1. Write equations expressing the equilibrium conditions. Remember that Fx = 0, Fy = 0 and τz = 0 are always separate equations; never add x- and y-components in a single equation. Also remember that when a force is represented in terms of its components, you can compute the torque of that force by finding the torque of each component separately, each with its appropriate lever arm and sign, and adding the results. This is often easier than determining the lever arm of the original force.

  10. Problem Solving Strategy • EXECUTEcontinued: • 2. You always need as many equations as you have unknowns. Depending on the number of unknowns, you may need to compute torques with respect to two or more axes to obtain enough equations. Often, there are several equally good sets of force and torque equations for a particular problem; there is usually no single “right” combination of equations. • EVALUATEyour answer: A useful way to check your results is to rewrite the second condition for equilibrium, τz = 0 using a different choice of origin. If you’ve done everything correctly, you’ll get the same answers using this new choice of origin as you did with your original choice.

  11. Solving rigid-body equilibrium problems • An auto magazine reports that a certain sports car has 53% of its weight on the front wheels and 47% on the rear, with a wheelbase of 2.46 m. This means that the total normal force on the front wheels is 0.53w and 0.47w on the rear. • How far in front of the rear axle is the car’s center of gravity?

  12. The ladder against a wall Sir Lancelot is trying to rescue the Lady Elayne from Castle Von Doom by climbing a uniform ladder that is 5.0 m long and weighs 180 N. Lancelot, who weights 800 N, stops a third of the way up on the ladder. The bottom of the ladder rests on a horizontal stone ledge and leans across the moat in equilibrium against a vertical wall that is frictionless. Find the normal and friction forces on the ladder at its base. Find the minimum coefficient of friction needed to prevent slipping. Find the contact force on the ladder at the base.

  13. Equilibrium while lifting weights • The forearm is in equilibrium under the action of the weight w of the dumbbell, the tension T in the tendon connected to the biceps muscle, and the force E exerted on the forearm by the upper arm at the elbow joint. The weight and angle are given; we want to find the tendon tension and the two components of force at the elbow. Neglect the weight of the forearm.

  14. Homework • Read 404 – 414 • 1, 5, 11, 17, 21

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