Problems 1-3 concern the following race. Consider three objects at the top of an

Question

Problems 1-3 concern the following race. Consider three objects at the top of an incline with = 12 and a height of 1.3 m: a sphere (I = 2 5mr2 ), a cylindrical shell (I = mr2 ), and a solid cylinder (I = 1 2mr2 ). Each object has a mass of 2 kg and a radius of 9 cm, and each is released from rest, then allowed to roll without slipping down the incline to the bottom.

1. What type(s) of energy do the three objects possess at the top of the incline? Find the total energy at the top of the incline.

2. At the bottom of the incline, each object will have both translational and rotational kinetic energy. a) What are the final velocities of each of the three objects? b) How much of the total energy is translational kinetic energy, and how much is rotational kinetic?

3. Now that you know the final velocity of each, solve for the time that each object will take each object to reach the bottom of the incline? Which one comes in first place? Second place? Third place?

Explanation / Answer

energy possecessed at top of incline is the potential energy

energy at top of incline irresespective of the object is mgh=2*10*1.3=26J

2.a velocity at bottom of incline is given as square root of 2gh/1+(k/r)2

for sphere k2 =2.5r2 v= 2.72 m/s

for cylinderical shell k2=r2 v= 3.6 m/s

for cylinder k2=2r2   v=1.414m/s

k.e/total energy=1/[1+(k/r)2]

for sphere this is 1/3.5=0.2857 is K.e and 1-0.2875=0.7125is rotational

for shell this 1/2=0.5 is kinetic and 1-0.5=0.5 is rotational

for solid cylinder =1/13=0.0769 is kinetic and 1-0.0769=0.9231 is rotational

t=square root of 2l(1+(k/r)2)/gsin12=2h(1+(k/r)2)/gsin212 t for sphere=4.76s

t for shell=3.60s

t for solid cylinder = 9.19s

sphere comes 1st shell 2nd and cylinder 3rd

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