Physics 211 11 Static Equilibrium and Elasticity Conditions

Physics 211 11: Static Equilibrium and Elasticity • Conditions for Static Equilibrium for Rigid Objects • Center of Gravity • Examples • Elastic Properties of Objects

Torque associated with a force Magnitude of the force times the perpendicular distance from the line of action of the force to the pivot point r d F

Conditions for Static Equilibrium for Rigid Objects The center of mass of an object does not accelerate F tot if the total force on the object is zero i. e. = 0 Û acm = 0 º TRANSLATIONAL EQUILIBRIUM An object will have zero angular acceleration if the total torque on the object is zero i. e t tot = 0 Û a = 0 º ROTATIONAL EQUILIBRIUM If the initial velocity of the center of mass is zero and the initial angular velocity is zero they will remain zero if F tot = 0 and a cm = 0 When this is so the object is said to be in STATIC EQUILIBRIUM

The torque of a force about any point on the line of action of that force is zero If a body is in translational equilibrium the net torque with respect to one point is the same with respect to ALL points in the body In particular if it is zero about one point It is zero about all points!

Total torque about point O is t O = r 1 ´ F 1 + r 2 ´ F 2 +L +r n ´ F n = å ri ´ Fi i total torque is taken about another point O ¢ t. O ¢ = å is (r i - r¢ ) ´ F i i where r ¢ is the displacement vector from O to O ¢ Þ t. O¢ = å( ri ´ Fi - r¢ ´ Fi) = i å( i Translational equilibrium Þ F 1 + F 2 +L +F n = t. O¢ = å( ri ´ F i ) = t. O i å å ri ´ F i ) - r¢ ´ Fi i Fi = 0 i

Center of Gravity 1 rcg = Wtot åW r = i i i mg r å åm g 1 i i i i Where the system is made up of discrete objects of mass mi and the acceleration of gravity at the location of these masses is gi. If the gravitational field strength is uniform i. e. does not vary, the center of gravity is at the same position as the center of mass

r cg = 1 W tot å i Wi ri = å å 1 mi g r i = i i å å 1 mi mi ri = r cm i i If system is an extended object the sums are replaced by integrals. If an object (extended or many discrete ones rigidly connected ) are suspended by an upward force in a homogenous gravitational field then when the object achieves static equilibrium Ftot = F up + Mtot g = 0 (no change of vcm from 0) Þ F up = -Mtot g t tot about center of mass must be zero (no change of acm from 0) t tot = text = 0 = r 1 ´ F up where r 1 is a displacement vector to the center of gravity Þ = center of mass r 1 is pointing towards the center of gravity and is parallel to the forc Þ Fup acts at a point vertically above the center of mass / gravity where the total weight vector acts

Elastic Properties of Objects States of Matter not resistant to forces gas liquid solid resistant to compression forces resistant to compression and shearing forces

compression shearing

Elastic Materials = Non rigid Elasticity is measured by the response of the material to an applied force.

Hookes Law F = -kd Restoring force is proportional to the displacement from the equilibrium position Displacement from the equilibrium position is proportional to the applied force

Stress = [Constant] x Strain

1 - Dimensional (Tension / Compression) F Dl ; = Y A l 0 3 - Dimensional DP = - B Y = Youngs Modulus (Tension / Compression ) DV ; B = Bulk Modulus V 0 3 - Dimensional (Shearing ) F Dx ; S = Shear Modulus = S A h