What is the Relationship Between Acceleration of a Block and a Pulley?

AI Thread Summary
The discussion focuses on the relationship between the acceleration of a block and a pulley in a frictionless system. It is established that the acceleration of the block (m1) is twice that of the pulley (m2), represented by the equation a1 = 2a2. Participants analyze the forces acting on both the block and the pulley, noting that the pulley is not in equilibrium due to the applied force and tension. A method involving the distances moved by the block and the pulley is suggested to derive the acceleration relationship. The conversation highlights a common misunderstanding in introductory mechanics regarding the equilibrium of the pulley.
imatreyu
Messages
79
Reaction score
0

Homework Statement



A horizontal force F is applied to a frictionless pulley of mass m2. The horizontal surface is smooth. Show that the acceleration of the block of mass m1 is twice the acceleration of the pulley.

LOOKS LIKE THIS: http://cnx.org/content/m14060/latest/npq1.gif
But WITHOUT block B and its string.

Homework Equations


F=ma


The Attempt at a Solution


I drew separate force diagrams for m1 (the block) and m2 (the pulley. In the x direction, the block is only being acted on by T1 going in the pos. x direction. In the x direction, the pulley is being acted on by 2T1 and F. 2T1 is going in the neg. x direction. F, the opposite.

I have to show that a1= 2a2

So:

The pulley is in equilibrium:
F-2T1 = 0
m2a2 - 2(m1a1)=0

. . .and I don't know where to go from here. . . .I can't eliminate mass. . .


Thank you in advance!
 
Physics news on Phys.org
It is easy to see that the pulley is moving. Let s_1 be the distance between pulley and A, s_2 be the distance between pulley and B. During the pull, s_2 stays constant. For s_1, it is decreasing right? But that section goes to the upper side of the pulley, so we can set 2s_1 equals also a constant. the reason of 2s_1 comes from the initial situation, we ignore the upper portion that is left of the originial position of A.

Hence 2s_1+s_2=constant, differentiate twice yields your desired result.
 
Oh. IDK why I thought the pulley was in equilibrium.

Thank you so much!
 
imatreyu said:
Oh. IDK why I thought the pulley was in equilibrium.

Thank you so much!

I made the same mistake as you when I was having a introductory mechanics class.
The method I present here is sometimes referred to no-stretch assumption. I don't know why the method is always not mentioned in the textbooks. Is it too obvious for the authors?
 
I suppose they think so! The textbook (College Physics, 3rd ed. Serway & Faughn) says nothing about using distances and time derivatives . . . xD

Thank you!
 
Last edited:

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
View full post »
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Similar threads

Solve Pulley-Mass System: m1=4.00 kg, m2=1.00 kg
Replies
2
Views
1K
Pulley system problem: 6 Pulleys and 1 Mass
Replies
5
Views
920
Olympiad dynamics problem with 3 masses and a pulley
Replies
17
Views
2K
Calculating acceleration of a prism and block connected to a wall through a rope and pulley
Replies
4
Views
391
Using Effective Mass to find the accelerations of 3 masses connected by pulleys
Replies
25
Views
3K
What are the accelerations of the blocks?
Replies
5
Views
2K
Angular Acceleration and Linear Acceleration of a Pulley
Replies
4
Views
2K
Acceleration of a block on a massive pulley
Replies
1
Views
2K
What if check: Am I calculating tension wrong?
Replies
3
Views
2K
Finding the acceleration of two masses on a pulley system
2
Replies
66
Views
7K

Hot Threads

Recent Insights

Back
Top