Slide 1 / 37 Rotational Motion
Slide 2 / 37 Angular Quantities An angle θ can be given by: where r is the radius and l is the arc length. This gives θ in radians. There are 360 in a circle or 2π radians. 360 = 2π rad 1 rad = 57.3 r θ r l
Slide 3 / 37 1 A 0.25 m arc in a circle of radius 5 m is being studied. How many radians is this?
Slide 4 / 37 2 A 0.25 m arc in a circle of radius 5 m is being studied. How many degrees is this?
Slide 5 / 37 Angular Quantities Just like velocity is the change in displacement over the change in time, angular velocity is defined as the change in angle over the change in time. Likewise, angular acceleration is the change in angular velocity over the change in time.
Slide 6 / 37 3 An object moves π/2 radians in 8 seconds. What is it's angular velocity?
Slide 7 / 37 4 An object starts from rest and accelerates to an angular velocity of 16 rad/s in 10 seconds. What is it's angular acceleration?
Slide 8 / 37 Angular Quantities Linear velocity can be related to angular velocity. The linear velocity of something moving in a circle would be the change in arc length over the change in time. Using θ = l/r, the change in length would be Δl = rδθ. And we already defined angular velocity as Δθ/Δt.
Slide 9 / 37 5 What is the linear speed of a child on a merry-goround of radius 3 m that has an angular velocity of 4 rad/s?
Slide 10 / 37 6 What is the angular velocity of an object traveling in a circle of radius 0.75 m with a linear speed of 3.5 m/s?
Slide 11 / 37 Angular Quantities Similarly, tangential acceleration can also be related to angular acceleration. But there is also radial acceleration, which we know to be v 2 /r but this can also be related to angular velocity.
Slide 12 / 37 Angular Quantities The linear acceleration is the vector sum of the angular and radial accelerations. a linear = a tan + a R a R a linear a tan
Slide 13 / 37 7 A child pushes a merry-go-round with radius of 2.5 m from rest to an angular velocity of 3 rad/s in 8 seconds. What is the angular acceleration?
Slide 14 / 37 8 What is the merry-go-round's tangential acceleration at that time?
Slide 15 / 37 9 What is the merry-go-round's radial acceleration at that time?
Slide 16 / 37 10 What is the merry-go-round's linear acceleration at that time?
Slide 17 / 37 Angular Quantities We know from circular motion that frequency is defined as the number of revolutions per second. So frequency can be given in terms of angular velocity. And angular velocity can be given in terms of frequency.
Slide 18 / 37 11 What is the frequency of the merry-go-round in the previous problem?
Slide 19 / 37 Kinematics Equations & Rolling Motion Just like the for kinematics equations for linear motion...
Slide 20 / 37 Kinematics Equations & Rolling Motion The same four exist for rolling motion...
Slide 21 / 37 12 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s 2 for 11 seconds. What is it's final angular velocity?
Slide 22 / 37 13 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s 2 for 11 seconds. What is it's final linear velocity?
Slide 23 / 37 14 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s 2 for 11 seconds. What is it's angular displacement during that time?
Slide 24 / 37 15 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s 2 for 11 seconds. How many revolutions did it make during that time?
Slide 25 / 37 16 A bicycle wheel with a radius of 0.3 m starts from rest and accelerates at a rate of 4.5 rad/s 2 for 11 seconds. What is it's linear displacement during that time?
Slide 26 / 37 17 A 50 cm diameter wheel accelerates from 5 revolutions per second to 7 revolutions per second in 8 seconds. What is the angular acceleration?
Slide 27 / 37 18 A 50 cm diameter wheel accelerates from 5 revolutions per second to 7 revolutions per second in 8 seconds. What is the angular displacement during that time?
Slide 28 / 37 19 A 50 cm diameter wheel accelerates from 5 revolutions per second to 7 revolutions per second in 8 seconds. How far will a point on the wheel have traveled during that time?
Slide 29 / 37 Torque Up to this point we described rotational motion without addressing the cause of rotational motion. The dynamics of rotational motion is analogous to the dynamics of linear motion, where force is replaced by torque. Torque is defined as force x lever arm (perpendicular).
Slide 30 / 37 20 A force of 300 N is applied to a crowbar 20 cm from the axis of rotation. Calculate the torque.
Slide 31 / 37 21 Now if that same force is applied to the same crow bar but now at a 25 o angle. Calculate the torque.
Slide 32 / 37 22 A student wants to balances two weights of 2 kg and 0.5 kg on a meter stick. How far does she have to put the fulcrum from the 2 kg weight?
Slide 33 / 37 In linear motion, a force causes an acceleration. In rotational motion, a torque causes an angular acceleration. F = ma Dynamics F = mrα since a = rα τ = mr 2 α since τ = Fr we can multiply both sides by r This is just for a single particle. For an entire body we take the sum of all the torques: Στ = Σ(mr 2 )α Where mr 2 is called the moment of inertia, I. Στ = Iα
Here are some moments of inertia for common objects. Slide 34 / 37 Moments of Inertia
Slide 35 / 37 Remember that kinetic energy is ½mv2. This is for an object that is undergoing translational motion. For an object undergoing rotational motion Kinetic Energy can be given as: KE = ½mv 2 Rotational Kinetic Energy KE = ½m(ωr) 2 since v = ωr KE = ½(mr 2 )ω 2 rotational KE = ½Iω 2
Slide 36 / 37 Work Since W = F dperpendicular... W = FΔl W = FrΔθ since θ = l/r W = τδθ since τ = Fr
Slide 37 / 37 Conservation of Angular Momentum Just like we have conservation of momentum and energy, we also can use the principle of conservation of momentum. Angular momentum is analogous to linear momentum. p = ma L = Iω The total angular momentum of an object is constant if the net torque on the object is zero.