Chapter 12 Equilibrium and Elasticity

Equilibrium and Elasticity - Definition - Requirements - Static equilibrium II. Center of gravity III. Elasticity - Tension and compression - Shearing - Hydraulic stress

I. Equilibrium - Definition: An object is in equilibrium if: - The linear momentum of its center of mass is constant. - Its angular momentum about its center of mass is constant Example: block resting on a table, hockey puck sliding across a frictionless surface with constant velocity, the rotating blades of a ceiling fan, the wheel of a bike traveling across a straight path at constant speed. Objects that are not moving either in TRANSLATION or ROTATION - Static equilibrium: Example: block resting on a table.

Stable static equilibrium: If a body returns to a state of static equilibrium after having been displaced from it by a force  marble at the bottom of a spherical bowl. Unstable static equilibrium: A small force can displace the body and end the equilibrium . (1) Torque about supporting edge by Fg=0 because line of action of Fg through rotation axis  domino in equilibrium. (1) Slight force ends equilibrium  line of action of Fg moves to one side of supporting edge  torque due to Fg increases domino rotation. (3) Not as unstable as (1)  to topple it one needs to rotate it through beyond balance position in (1).

- Requirements of equilibrium: . - Requirements of equilibrium: Balance of forces  translational equilibrium Balance of torques  rotational equilibrium

- Equilibrium: Vector sum of all external torques that act on the body, measured about any possible point must be zero. Vector sum of all external forces that act on body must be zero. Balance of forces  Fnet,x = Fnet,y = Fnet,z =0 Balance of torques  τnet,x = τnet,y = τnet,z =0

- This course initial assumption: II. Center of gravity Gravitational force on extended body  vector sum of the gravitational forces acting on the individual body’s elements (atoms) . cog = Body’s point where the gravitational force “effectively” acts. The center of gravity is at the center of mass. - This course initial assumption: If g is the same for all elements of a body, then the body’s Center Of Gravity (COG) is coincident with the body’s Center Of Mass (COM). Assumption valid for every day objects  “g” varies only slightly along Earth’s surface and decreases in magnitude slightly with altitude.

Each force Fgi produces a torque τi on the element of mass about the origin O, with moment arm xi.

A baseball player holds a 36-oz bat (weight = 10 A baseball player holds a 36-oz bat (weight = 10.0 N) with one hand at the point O . The bat is in equilibrium. The weight of the bat acts along a line 60.0 cm to the right of O. Determine the force and the torque exerted by the player on the bat around an axis through O.

A uniform beam of mass mb and length ℓ supports blocks with masses m1 and m2 at two positions. The beam rests on two knife edges. For what value of x will the beam be balanced at P such that the normal force at O is zero?

A circular pizza of radius R has a circular piece of radius R/2 removed from one side as shown in Figure. The center of gravity has moved from C to C’ along the x axis. Show that the distance from C to C’ is R/6. Assume the thickness and density of the pizza are uniform throughout.

Pat builds a track for his model car out of wood, as in Figure Pat builds a track for his model car out of wood, as in Figure. The track is 5.00 cm wide, 1.00 m high and 3.00 m long and is solid. The runway is cut such that it forms a parabola with the equation Locate the horizontal coordinate of the center of gravity of this track.

Find the mass m of the counterweight needed to balance the 1 500-kg truck on the incline shown in Figure. Assume all pulleys are frictionless and massless.

A 20.0-kg floodlight in a park is supported at the end of a horizontal beam of negligible mass that is hinged to a pole, as shown in Figure. A cable at an angle of 30.0° with the beam helps to support the light. Find (a) the tension in the cable and (b) the horizontal and vertical forces exerted on the beam by the pole.

States of Matter Matter is characterised as being solid, liquid or gas Solids can be thought of as crystalline or amorphous For a single substance it is normally the case that solid state occurs at lower temperatures than liquid state and liquid state occurs at lower temperatures than gaseous state

Solids and Fluids Solids are what we have assumed all objects are up to this point (rigid, compact, unchanging, simple shapes) We will now look at bulk properties of particular solids and also at fluid Fluids include both liquids and gases fluids assume the shape of their container

Deformation of Solids All states of matter (S,L,G) can be deformed it is possible to change the shape and volume of solids and the volumes of liquids and gases Any external force will deform matter for solids the deformation is usually small in relation to its overall size when ‘everyday’ forces are applied

Stress and Strain When force is removed, the object will usually return to its original shape and size Matter is elastic We characterise the elastic properties of solids in terms of stress (amount of force applied) and strain (extent of deformation) that occur. The amount of stress required to produce a particular amount of strain is a constant for a particular material this constant is called the elastic modulus

Young’s Modulus (length elasticity) F creates a tensile stress of F/A. Units of tensile stress are Pascal 1 Pa = 1 N/m2 Change in length created by stress is DL/Lo (no unit) This quantity is the tensile strain F A Lo

Young’s modulus Young’s modulus is the ratio of tensile stress to tensile strain. Y has units of Pa Young’s modulus applies to rod or wire under tension (stretching) or compression

What is happening in detail? Bonds between atoms are compressed or put in tension

Elastic behaviour Notice that the Stress is proportional to the Strain This is similar to the relation we had between spring force and its extension |F| = kx We can identify k with YA/Lo F = kx so F/A = kx/A=YDL/Lo

Limits of elastic behaviour

Shear Modulus Shear modulus characterises a body’s deformation under a sideways (tangential) force A h F Dx

Bulk Modulus Response of body to uniform squeezing Bulk modulus is ratio of the change in the normal force per unit area to the relative volume change F

Values for Y, S and B Y (N/m2) S (N/m2) B (N/m2) Al 7.0 x 1010 Water - -* 0.21 x 1010 Tungsten 35 x 1010 14 x 1010 20 x 1010 Glass 7 x 1010 3 x 1010 5.2 x 1010 Note that liquids do not have Y defined. S for liquids is called viscosity. In liquids and gases S and B are strongly dependent on temperature (more later)

Example - Young’s modulus How much force must be applied to a 1 m long steel rod that has end area 1 cm2 to fit it inside a 0.999 m long case? (Ysteel=20x1010N/m2) 1 cm2

Density Density of a pure solid, liquid, or gas is its mass per unit volume Density r  m/V (units are kg/m3)

Pressure Pressure is the force per unit area P  F/A Pressure at any point can be measured by placing a plate of known area in (e.g.) a liquid and measuring the force (compression) on a spring attached to it.