Rolling Cylinders: Find Final Velocity & Height

In summary, the final linear velocity of the (b) thin and (a) solid cylinder is 3.13 m/s and the height of the other object is 0.25 m at the time it reaches the ground.
  • #1
PrideofPhilly
37
0

Homework Statement



A thin cylindrical shell and a solid cylinder have the same mass and radius. The two are released side by side and roll down, without slipping, from the top of an inclined plane that is 1 m above the ground. The acceleration of gravity is 9.8 m/s2.

Find the final linear velocity of the (b) thin and (a) solid cylinder.

(c) When the first object reaches the bottom, what is the height above the ground of the other object?

Homework Equations



E = mgh + 1/2mv^2 + 1/2Iw^2

w = v/r

I = mr^2

I = 1/2mr^2

The Attempt at a Solution



(a) vf (solid) = (4gh/3)^1/2 = 3.61 m/s

(b) vf (thin) = (gh)^1/2 = 3.13 m/s

(c)?

I don't know where to start for part c.
 
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  • #2
Solve for the time it takes for the first cylinder to reach the ground, and plug that time back into the equation of motion of the other cylinder to find its height at that time.
 
  • #3
I still don't know which equation to use!

Is it:

v = -1/2gt
t = 0.368

and then what?

or am I still off?
 
  • #4
PrideofPhilly said:
I still don't know which equation to use!

Is it:
v = -1/2gt
t = 0.368

and then what?
or am I still off?

Consider

y = 1/2*a*t2
 
  • #5
well then:

y = (1/2)(9.8)(0.368)^2
y = 0.66 m

BUT this answer is wrong!

The answer is 0.25 m.

Am I using the right acceleration and time?
 
  • #6
PrideofPhilly said:
well then:

y = (1/2)(9.8)(0.368)^2
y = 0.66 m

BUT this answer is wrong!

The answer is 0.25 m.

Am I using the right acceleration and time?

Don't you want to consider the time of the faster, in the equation of the distance the slower will have gone and then take the difference?
 
  • #7
LowlyPion said:
Don't you want to consider the time of the faster, in the equation of the distance the slower will have gone and then take the difference?

I'm sorry but what does this mean?

I don't understand what you just said.
 
  • #8
SOME BODY PLEASE HELP! I'm so confused on this problem!
 
  • #9
PrideofPhilly said:

The Attempt at a Solution


(a) vf (solid) = (4gh/3)^1/2 = 3.61 m/s
(b) vf (thin) = (gh)^1/2 = 3.13 m/s
(c)?

You've found that

v_solid2 = 4/3*gh
v_thin2 = gh

Consider also then that

v2 = 2*a*x

If you explore the relationship of the ratio of the velocity2 one to the other you will have a ratio of the accelerations don't you?

Armed with that you also know that

x = 1/2*a*t2

What happens then when you plug in the acceleration of the slower, to the equation of the faster? For the same t2 what will the drop have been?

All you need to do then is determine how much further the slower has to go ... the answer to part C.
 

1. What is the equation for finding the final velocity of a rolling cylinder?

The equation for finding the final velocity of a rolling cylinder is v = ωr, where v is the final velocity, ω is the angular velocity, and r is the radius of the cylinder.

2. How do you find the height of a rolling cylinder?

To find the height of a rolling cylinder, you can use the equation h = (v2)/2g, where h is the height, v is the final velocity, and g is the acceleration due to gravity.

3. Can the final velocity of a rolling cylinder ever be greater than the initial velocity?

No, the final velocity of a rolling cylinder can never be greater than the initial velocity. This is because some of the initial kinetic energy is converted into potential energy as the cylinder reaches its maximum height.

4. How does the mass of the cylinder affect the final velocity and height?

The mass of the cylinder does not affect the final velocity, as it is not included in the equations for finding the final velocity or height. However, a heavier cylinder will have a greater potential energy at its maximum height compared to a lighter cylinder.

5. What are the assumptions made when calculating the final velocity and height of a rolling cylinder?

The assumptions made when calculating the final velocity and height of a rolling cylinder include: the surface is frictionless, the cylinder is rolling without slipping, and there is no air resistance. These assumptions may not hold true in real-world scenarios and can affect the accuracy of the calculated values.

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