L21-s1,11 Physics 114 – Lecture 21 §7.6 Inelastic Collisions In inelastic collisions KE is not conserved, which is equivalent to saying that mechanical energy is not conserved Collisions where the two bodies stick together after the collision are known as completely inelastic collisions Study Example 7.9 – the energy lost during a collision between two railroad cars We now study the ballistic pendulum

L21-s2,11 Physics 114 – Lecture 21 Example 7.10 – the ballistic pendulum Momentum conservation in collision → mv = (m + M)V ´ Mechanical energy conservation after collision → ½ (m + M)V ´2 = (m + M)gh h = L – L cosθ = L (1 – cosθ) M m v V M = 0 m + M V´V´ h Lθ

L21-s3,11 Physics 114 – Lecture 21 §7.8 Center of Mass (CM) So far we have considered the motion of point particles, that is the motion of a body where its position can be specified precisely What coordinates do we use when we deal with an extended body, c.f., races in athletics, etc.? It turns out that, by defining a quantity known as the Center of Mass (CM), all of our equations in Newton’s Mechanics remain valid provided that we consider the motion of the center of mass of that body

L21-s4,11 Physics 114 – Lecture 21 Definition of Center of Mass Location of the center of mass (CM) is defined to be and, generalizing to two dimensions, we have for y CM, x0 x1x1 x2x2 x3x3 x4x4 x 4 … m1m1 m2m2 m3m3 m4m4 m 4 …

L21-s5,11 Physics 114 – Lecture 21 Chapter 8 Rotational Motion §8.1 Angular Quantities So far we have considered point particles where no complications occurred regarding its rotation Once we consider the motion of an extended body we need to consider its rotational motion First we need to remind ourselves about angular quantities, where θ is expressed in radians or s = rθ Study Example 8.1 r θ s (along arc)

L21-s6,11 Physics 114 – Lecture 21 Rotational motion of a wheel The wheel rotates so that a point on the rim moves from an initial position, θ 1, to a final position, θ 2 Δ θ = θ 2 - θ 1 Define the average angular velocity, As before the instantaneous angular velocity is then defined as, θ1θ1 θ2θ2

L21-s7,11 Physics 114 – Lecture 21 We define, similarly, the average angular acceleration, and the instantaneous angular acceleration, Each point on the rim of the wheel has a linear velocity, ΔsΔs Δθ r

L21-s8,11 Physics 114 – Lecture 21 Note that the expression, v = r ω is valid for the linear velocity at any radius on the wheel Likewise this expression holds for any rotating body, where r is the distance from the axis of rotation to the point at a distance, r, perpendicular to this axis and provided also that the object is a rigid body, that is that all points in the body rotate with the same angular velocity, ω Study Examples, 8.3, 8.4 and 8.5

L21-s9,11 Physics 114 – Lecture 21 §8.2 Constant Angular Acceleration Since the definitions of ω and α from θ and t correspond exactly to the definitions of v and a from x and t, we have a similar set of relations between ω, α, θ and t

L21-s10,11 Physics 114 – Lecture 21 Kinematic Formulae for Rotational Motion Linear Angular v = v 0 + a tω = ω 0 + α t x = x 0 + v 0 t + ½ a t 2 θ = θ 0 + ω 0 t + ½ α t 2 v 2 = v a (x - x 0 ) ω 2 = ω α (θ - θ 0 ) Study Example 8.6

L21-s11,11 Physics 114 – Lecture 21 §8.3 Rolling Motion (Without Slipping) v = r ω and a = r α Study Example 8.7