Descending mass on rope attached to wheel

In summary: You can start by making a rough estimate of the answer. Here, the person falls about 12 meters, so the time should be about the square root of 12/9.8 seconds, or about 1.1 seconds - less than half of the more accurate result. This is because the wheel is turning, and the rope is getting shorter as the person falls. In the first half of the fall, the rope wraps around the wheel, so the person doesn't fall as fast; in the second half, the rope unwraps, so the person falls faster. This makes the average speed slower than the final speed. There's also a bit of a fudge factor because the rope is attached to
  • #1
ConorDMK
25
0

Homework Statement


A physics student of mass 57.0 kg is standing at the edge of the flat roof of a building, 12.0 m above the sidewalk. An unfriendly dog is running across the roof toward her. Next to her is a large wheel mounted on a horizontal axle at its center. The wheel, used to lift objects from the ground to the roof, has a light crank attached to it and a light rope wrapped around it; the free end of the rope hangs over the edge of the roof. The student grabs the end of the rope and steps off the roof.

a) If the wheel has radius 0.300 m and a moment of inertia of 9.60 kg⋅m2 for rotation about the axle, how long does it take her to reach the sidewalk? Ignore friction.

b) How fast will she be moving just before she lands? (I have not done anything for part b yet)

Homework Equations


I don't know if these are relevant, and I am sure I am missing some.

F=ma

ω=θ/t

v=ωr

θ=h/r

The Attempt at a Solution


As the person descends, there is a constant tension in the rope so the force acting on the person is
ΣF=ma=T-mg (Taking up as positive)

The height of the building is equal to the amount of rope required to descend, so the angle that the wheel must rotate through is
θ=h/r

and so the angular velocity of the wheel is
ω=h/(rt)

But this is where is cannot figure out where to go from here.
I did try to solve this with energy, but it kept giving me an incorrect answer.
The moment of inertia does suggest energy method though, but I cannot get it to work properly.
 
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  • #2
ConorDMK said:
As the person descends, there is a constant tension in the rope so the force acting on the person is
ΣF=ma=T-mg (Taking up as positive)
OK, you've applied Newton's 2nd law to the person. Good!

Now do the same thing for the wheel. Then you'll be able to solve for the acceleration.
 
  • #3
Doc Al said:
OK, you've applied Newton's 2nd law to the person. Good!

Now do the same thing for the wheel. Then you'll be able to solve for the acceleration.

Do you mean apply the rotational form of Newton's 2nd law? τ=Iα
 
  • #4
ConorDMK said:
Do you mean apply the rotational form of Newton's 2nd law? τ=Iα
Absolutely.
 
  • #5
Doc Al said:
Absolutely.

But we haven't been taught that yet, so I wouldn't know how torque works.
 
  • #6
ConorDMK said:
But we haven't been taught that yet, so I wouldn't know how torque works.
No problem. Use conservation of energy.

You tried it and got the wrong answer, so show what you did. (One key is to properly relate the linear speed of the falling person to the angular speed of the turning wheel.)
 
  • #7
Doc Al said:
No problem. Use conservation of energy.

You tried it and got the wrong answer, so show what you did. (One key is to properly relate the linear speed of the falling person to the angular speed of the turning wheel.)

Energy before:
Etotal=mgh

Energy after:
Etotal=0.5mv2+0.5Iω2

using ω=h/(rt) and v=2h/t

mgh=0.5m(2h/t)2+0.5I(h/(rt))2

t2=(2mh2+0.5I(h/r)2)/(mgh)

t=√[(2mh2+0.5I(h/r)2)/(mgh)]

Putting in the values

t=√[(2(57kg)(12m)2+0.5I(12m/0.3m)2)/((57kg)(9.8N/kg)(12m))]

t=1.90s to 2 decimal places (1.89596... s)

I don't know if this would be the answer if I had of used τ=Iα.
 
  • #8
ConorDMK said:
Energy before:
Etotal=mgh

Energy after:
Etotal=0.5mv2+0.5Iω2
This is good.

ConorDMK said:
using ω=h/(rt) and v=2h/t
Why is there a factor of 2 in the expression for v, but not in the expression for ω?
 
  • #9
Doc Al said:
This is good.Why is there a factor of 2 in the expression for v, but not in the expression for ω?

suvat: s=t(v+u)/2 as u=0 for both
v=2s/t

so v=2h/t and ω=2h/(rt)

Correct? or should they be without the 2?
 
  • #10
ConorDMK said:
suvat: s=t(v+u)/2 as u=0 for both
v=2s/t

so v=2h/t and ω=2h/(rt)

Correct?
Correct!

ConorDMK said:
or should they be without the 2?
Nope, the 2 is correct.
 
  • #11
Doc Al said:
Correct!Nope, the 2 is correct.

so if I continue with mgh=0.5m(2h/t)2+0.5I(2h/(rt))2 I should get the correct answer?
 
  • #12
ConorDMK said:
so if I continue with mgh=0.5m(2h/t)2+0.5I(2h/(rt))2 I should get the correct answer?
Yes. Just solve for t.
 
  • #13
Doc Al said:
Yes. Just solve for t.
t=√[(2(57kg)(12m)+2(9.6kgm2)(12m/0.3m)2)/((57kg)(9.8N/kg)(12m))]

t=2.65s
 
  • #14
ConorDMK said:
t=√[(2(57kg)(12m)+2(9.6kgm2)(12m/0.3m)2)/((57kg)(9.8N/kg)(12m))]

t=2.65s
Looks good.
 

1. How does the mass of an object on a rope attached to a wheel affect its descent?

The mass of an object on a rope attached to a wheel will affect its descent by increasing the force of gravity acting on the object. This means that the object will fall faster and with more acceleration compared to a lighter object.

2. Does the length of the rope affect the descent of the mass?

Yes, the length of the rope does affect the descent of the mass. A longer rope will result in a longer distance for the mass to fall, while a shorter rope will result in a shorter distance. This is due to the increase or decrease in the force of gravity acting on the object.

3. Will the shape of the wheel impact the descent of the mass?

Yes, the shape of the wheel can impact the descent of the mass. A larger wheel will result in a longer distance for the mass to fall, while a smaller wheel will result in a shorter distance. This is due to the change in circumference and the speed at which the wheel rotates.

4. How does friction play a role in the descent of the mass on a rope attached to a wheel?

Friction between the rope and the wheel can slow down the descent of the mass. This is because friction creates resistance, which counteracts the force of gravity and decreases the speed at which the object falls.

5. Is the descent of the mass affected by the material of the rope?

Yes, the material of the rope can affect the descent of the mass. A heavier or thicker rope will result in a slower descent due to the increased friction between the rope and the wheel. A lighter or thinner rope will result in a faster descent due to less friction.

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