Finding the Time for Mass 1 to Hit the Floor in a Rotating Rigid Body System

In summary, the Homework Statement states that two blocks are connected by a massless rope that passes over a pulley. The pulley has a mass of 2.4 kg, and as the pulley turns, friction at the axle exerts a torque of magnitude 0.54 Nm.
  • #1
hyddro
74
2

Homework Statement


The two blocks in the figure are connected by a massless rope that passes over a pulley. The pulley is 14cm in diameter and has a mass of 2.4 kg. As the pulley turns, friction at the axle exerts a torque of magnitude 0.54 Nm .

12.P70.jpg


Homework Equations


so m1 = 4.0kg
m2= 2.0 kg
M = 2.4kg
r= 0.07m
Tf (torque due to friction) = 0.54 Nm
ƩFy (for mass 1)= T2 - m1*g
m1*ay1 +m1*g= T2 so T2 = m1(ay1+g) this becomes T2= m1(ay + g)

ƩFy (for mass 2)= T1 - m2*g
m1*ay2 + m2*g = T1 so T1 = m2(ay2 + g)
but since ay1 = -ay2 = ay this becomes T1 = m2(g - ay)

Ʃτ= T2*R - T1*R - Tf

The Attempt at a Solution



Using what i put up there i get the following formula

τnet = R(T2 - T1) - Tf = R(m1(ay+g) - m2(g-ay)) - Tf ... equation 1
since τ= Iα, I= 1/2 MR^2 and since α= -ay/R

equation 1 becomes..

1/2MR^2 * (-ay)R = R(m1(ay+g) - m2(g-ay)) - Tf ... solving for ay with the given data i found ay=-1.65 and thus using the kinematic equation i found Δt= 1.55s but this is wrong.. :( i would appreciate if anyone points out my error or mistake, thanks
EDIT: sorry about this, they are asking us to find the time it takes for mass 1 to hit the floor, starting at rest.
 
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  • #2
Well, you never said what the actual question is. Looks like you are to figure out how long it takes the 4 kg mass to hit the floor? And everything is initially at rest?

I agree with the 1.65 m/s2 acceleration, so your mistake is probably in applying the kinematic equations. Can you show that work?

p.s. It's kind of confusing that you chose T2 for the rope holding m1, and T1 for the rope holding m2. Nonetheless you did find the correct acceleration.
 
  • #3
thanks! and sorry about that! Yes it is kinda confusing but I am happy to know that someone else got the same acceleration. Well, I tried applying the kinematic formula:

yf = yi + Vi*t + 1/2 a t^2 ... t = delta t... and i chose the initial point to be 0, so the final point would be -2m since it goes down... also, the initial velocity is zero since it starts from rest... so..

-2.0m = 0 + 0 * t + 1/2 * (-1.65m/s^2) * t^2
thus...

-2.0 m / -1.65 m/s^2 = t^2... applying square root i get 1.1s...which seems to be the right answer... what the hell was i doing wrong?? oh wow i think i was not dividing properly... lol so all this time i actually had the right answer before my eyes but never realized that i was making a silly mistake... my mistake was applying the kinematic formula.. well thanks for your help have a nice day :)
 
  • #4
[STRIKE]I get an acceleration of around 3.7 m/s and around 0.7 s.[/STRIKE] The only thing I can spot is that you multiply with R on the left-hand side of you last equation in your first post instead of dividing, but that just looks like a typo.

Edit: Just for the record, I made a sign error. Doing it properly I too get 1.7 m/s2 and 1.1 s.
 
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  • #5
I get 1.1 s as well. :smile: Be careful, it actually drops 1.0 m in 1.1 s, not 2.0 m.
 
  • #6
wait, how come it only drops 1 meter? i thought it was going to hit the ground? are you saying that it doesn't actually hit the floor?... how come i got the same answer then? sorry but that made me confused...

EDIT! OH OH! sorry i got 2m because i multiplied both sides of the last equation by 2 so i get rid of the 1/2 on the right side of the equation. ofc its going to drop 1m cause that's the height of the mass1, i think i was looking at a different problem... lol thanks! you guys rock!
 
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1. What is a rigid body?

A rigid body is an object that does not deform or change shape when subjected to external forces. In other words, the distances between all points on the object remain constant, and the object maintains its original shape and size.

2. What is the difference between rotation and translation?

Rotation is the movement of an object around a fixed axis, while translation is the movement of the entire object in a straight line without any rotation. In other words, rotation changes the orientation of an object, while translation changes its position.

3. What is the moment of inertia?

The moment of inertia is a measure of an object's resistance to rotational motion. It depends on the mass distribution of the object and the axis of rotation. The larger the moment of inertia, the more force is required to rotate the object.

4. How is the angular velocity of a rigid body calculated?

The angular velocity of a rigid body is calculated by dividing the change in angular displacement by the change in time. It is represented by the Greek letter omega (ω) and is measured in radians per second.

5. How does the rotation of a rigid body affect its kinetic energy?

The rotation of a rigid body can affect its kinetic energy by changing its moment of inertia. As the moment of inertia increases, the rotational kinetic energy also increases, and vice versa. This relationship is described by the equation: KER = (1/2) * I * ω^2, where I is the moment of inertia and ω is the angular velocity.

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