LECTURE 22 EQUILIBRIUM. Instructor: Kazumi Tolich

Transcription

1 LECTURE 22 EQUILIBRIUM Instructor: Kazumi Tolich

2 Lecture 22 2 Reading chapter 11-3 to 11-4 Static equilibrium Center of mass and balance

3 Static equilibrium 3 If a rigid object is in equilibrium (constant velocity and constant angular velocity), F $%& = 0 and τ $%& = 0 For a rigid object to be in static equilibrium (motionless), the linear and angular velocities of the object also must be zero as well as the above conditions being met.

4 Static equilibrium problem solving strategy 4 1. Identify all the forces on a system. 2. The net force on the system in each direction must be zero. 3. Pick an axis. 4. The net torque on the system about that axis must be zero.

5 Quiz: 1 A uniform meter-stick of a mass m is held perpendicular to a vertical wall by a string of a length L going from the wall to the far end of the stick. Around which axis should you calculate the torque in order to calculate the tension in the string? Wall String Meter stick

6 Quiz: 22-1 answer 6 If we place an axis anywhere on the string, we will eliminate torque due to tension, so we cannot solve for tension. Since we have no information on the normal force or the friction by the wall on the meter-stick, by placing the axis there, we can eliminate the torques due to these unknown forces. Wall f /0 String T /3 In general, your calculation can be simplified by choosing an axis that goes through the line of action of the force that you have the least information about. N /0 Meter stick W /5 = mg

7 Example: 1 7 A meter stick with a mass m = 0.16 kg is held perpendicular to a vertical wall by a string of a length L = 2.5 m going from the wall to the far end of the stick. Find the tension in the string. String L Wall Meter stick

8 Quiz: 2 8 Suppose a ladder of weight W 95 on a rough floor is leaning against a frictionless wall. The top of the ladder is height h above the floor, and the bottom of the ladder is a distance 2r away from the wall. An object of weight W >5 is added to the ladder at a distance x from the bottom of the ladder. If you want to apply Newton s 2 nd law for rotation, about which axis on the ladder do you want to calculate the net torque to minimize the number of terms in the resulting equation? N 9: W W >5 95 N 90 h f ;,9: r x r

9 Quiz: 22-2 answer 9 With the chosen axis, N 9: or f ;,9: does not contribute to the net torque as their lever arms are both zero. N 90 W W >5 95 h N 9: f ;,9: r x r

10 Example: 2/Demo: 2 10 Suppose a ladder of weight W 95 is leaning against a frictionless wall on a rough floor. The top of the ladder is height h above the floor, and the bottom of the ladder is a distance 2r away from the wall. An object of weight W >5 is added to the ladder at a distance x from the bottom of the ladder. What are the magnitudes of each of the forces on the ladder? N 9: W W >5 95 N 90 h f ;,9: r x r

11 Example: 3/Demo: 2 11 A toy car of mass m A crosses a bridge of length L and mass m B. The bridge is supported at both ends. What is the support force at both ends in terms of variables given as function of the position of the car from the left end of the bridge, x? x L m A m B

12 Equilibrium of a suspended object 12 If you allow an arbitrarily shaped object to hang freely, its center of mass is directly below the suspension point. The torque due to gravity is zero when the center of mass is directly below the suspension point.

13 Equilibrium on a surface 13 An object is in equilibrium if its center of mass is directly above the base on which it is supported. If not, it will tip over. The center of mass of the system (bottle plus holder) is directly over the support point.

14 Carrying water 14 Why is it easier to carry the same amount of water in two buckets, one in each hand, than in a single bucket in one hand? With two buckets, the center of mass will be in the center of the support base provided by one s feet, so there is no need to lean. Good posture places the upper body s center of mass over the pivots in the hips. Poor posture requires exertion by the back muscles to counteract the torque produced around the pivot by the upper body s weight. CM CM CM

15 Forces and torques in muscles and joints 15 Most skeletal muscles exert much larger forces within the body than the limbs apply to the outside world. Most muscles are attached to bones via tendons close to joints, causing these systems to have mechanical advantages much less than one.

16 Quiz: 3 16 An object is made by hanging a ball of mass M from one end of a plank having the same mass and length L. The object is then pivoted at a point a distance L 4 from the end of the plank supporting the ball, as shown below. Is the object balanced? A. Yes B. No

17 Quiz: 22-3 answer 17 Yes If the net torque on the object about the pivot is zero, the object would balance. τ = Mg G H Mg G H = 0 CM Plank