Rotational Kinetic Energy Which would have a greater Rotational Kinetic Energy? Both scenarios have a total of 8 tennis balls, all with a mass m, being spun at a constant rotational speed of ω, in a circle by length of string R. A. B. C. Both the same D. Can’t be determined

Rotational Kinetic Energy Ekr (J) When an object rotates it has Ekr. Objects can have both linear & rotational kinetic energy Ekr due to rotation Ek of linear motion(center-of-mass) Assuming conservation of Mechanical energy holds, then: d θ°

See weebly for animations http://wsapphysics.weebly.com/rotational-inertia-e--p.html

Like a Rolling Disk E#1 A 1.20 kg disk with a radius 0f 10.0 cm rolls without slipping. The linear speed of the disk is v = 1.41 m/s. (a) Find the translational kinetic energy. (b) Find the rotational kinetic energy. (c) Find the total kinetic energy.

Check for Understanding

8.4 Check for Understanding

Rolling Race (Hoop vs Cylinder) A hoop and a cylinder of equal mass roll down a ramp with height h. Which has greatest KE at bottom? A) Hoop B) Same C) Cylinder 20% 50% 30% I = MR2 I = ½ MR2

Rolling Down an Incline

E#2 Biscuit wheel will reach the bottom of the slope first. The ring and biscuit wheels have identical mass and radius. Both ramps are identical. Both wheels are released at the same time. Which wheel will reach the bottom of the slope first? Explain your answer E#2 Biscuit wheel will reach the bottom of the slope first. ring The inertia of the ring wheel is more that of the inertia of the biscuit wheel so the ring wheel is harder to get started. The biscuit wheel having a lower inertia is easier to accelerate. Since the biscuit has less rotational inertia then it will also have less Rotational Kinetic Energy at the bottom of the hill compared to the ring (Krot=1/2Iw2) Since the biscuit has less Krot it has more K – since both start with the same Ug and Ug = K + Krot. biscuit

Compare Heights A ball is released from rest on a no-slip surface, as shown. After reaching the lowest point, it begins to rise again on a frictionless surface. When the ball reaches its maximum height on the frictionless surface, it is higher, lower, or the same height as its release point? The ball is not spinning when released, but will be spinning when it reaches maximum height on the other side, so less of its energy will be in the form of gravitational potential energy. Therefore, it will reach a lower height.

E#3 2 m A truck wheel has a mass of 65 kg and a radius of 0.56 m. It is rolling down a 2.0 m high slope. The truck wheel gains 540 J of rotational energy by the time it reaches the bottom of the slope. Show the wheel's translational velocity at the bottom of the slope is 4.8 m s-1. Show the wheels angular velocity is 8.6 s-1 at the bottom of the slope. Calculate the rotational inertia of the truck wheel. 2 m E#3

Show the wheel's translational velocity at the bottom of the slope is 4.8 ms-1.

Show the wheels angular velocity is 8.6 s-1. Calculate the rotational inertia of the truck wheel.

Assume all spheres must be the same, i.e. solid and round. Say Galileo tested his free fall theory by rolling different diameter solid spheres down a slope, instead of dropping them off a ledge. Prove that his free fall theory is correct. Galileo Galilei Assume all spheres must be the same, i.e. solid and round.

Spinning Wheel E#4 A block of mass m is attached to a string that is wrapped around the circumference of a wheel of radius R and moment of inertia I, initially rotating with angular velocity w that causes the block to rise with speed v . The wheel rotates freely about its axis and the string does not slip. To what height h does the block rise?

A Bowling Ball A bowling ball that has an 11 cm radius and a 7.2 kg mass is rolling without slipping at 2.0 m/s on a horizontal ball return. It continues to roll without slipping up a hill to a height h before momentarily coming to rest and then rolling back down the hill. Model the bowling ball as a uniform sphere and calculate h. E#5

Betty-Sue and Bubba decide to roll different objects down a long sloping, smooth driveway.  The driveway can be treated as a ramp (see animation) and it has a 15 degree angle to the horizontal.  Their 1st object is a ball of 438grams and a diameter of 19.5cm that rolls from rest 5.9m along the driveway. (a)  Calculate the angular displacement of the ball as it rolls down the ramp. radius = = 0.0975m = 60.512… ≈ 61 rad (2 SF) 5.9m 15° E#6 Applet link

(b) The ball takes 3. 19s to reach the bottom of the driveway (b)  The ball takes 3.19s to reach the bottom of the driveway. Calculate the final angular velocity and the angular acceleration of the ball.

(c)  Calculate the rotational kinetic energy of the ball at the base of the ramp. (hint: use other energies to find your answer)

(d)  What assumption(s) did you make in answering the previous question? Assumption: that all of the gravitational potential energy at the top of the hill was converted to linear kinetic and rotational kinetic energies at the base of the drive.  Or: no energy was “lost” to heat or sound as the ball rolled down the drive.

(e)  Calculate the rotational inertia of the ball. (f)  Calculate the angular momentum of the ball at the base of the ramp.

(g) Later, Bubba lets 2 different masses go at the same time (g)  Later, Bubba lets 2 different masses go at the same time.  He chooses a disk and a ring of the same mass and radius .  Explain which reaches the bottom of the ramp first and why. The disk will reach the bottom before the ring. Since they have the same mass but different mass distribution, they have different rotational inertias while having the same total energy (EP at the top of the hill). The ring has a larger rotational inertia, I, than the disk, since more of its mass is farther from the centre. Thus the ring will have a larger portion of its original gravitational potential energy converted into rotational kinetic energy than the disk.  This leaves a small portion of energy for linear kinetic, thus a smaller linear speed for the ring.