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Excerpt
hiddentrue

A mass falling while attached to a massive pulley.

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Composition Setup
A bucket for collecting water from a well is suspended by a rope which is wound around a pulley. The empty bucket has a mass of 2.0 kg, and the pulley is essentially a uniform cylinder of mass 3.0 kg on a frictionless axle. Suppose a person drops the bucket (from rest) into the well. 
 Deck of Cards
idbigdeck
 Card
labelPart A
Part A
What is the bucket's acceleration as it falls?
Solution 
We will consider two different methods to obtain the solution.
 Deck of Cards
idlittledeck
 Card
labelMethod 1: Dynamics
Method 1: Dynamics
 Toggle Cloak
idsysa1
 Systems: 
 Cloak
idsysa1
The pulley and the bucket are treated as separate objects. The bucket can be treated as a , but the pulley must be treated as a . 
 Toggle Cloak
idinta1
 Interactions: 
 Cloak
idinta1
The pulley and the bucket are each subject to external influences from the rope () and from the earth (gravity). The pulley is also subject to a  from the axle, but we will choose the axle as the axis of rotation so the axle  produces no torque (it has zero ). 
 Toggle Cloak
idmoda1
 Model: 
 Cloak
idmoda1
 and 
 Toggle Cloak
idappa1
 Approach: 
 Cloak
idappa1
 Toggle Cloak
idinta2
 Interactions: 
 Cloak
idinta2
The system is subject to external forces from the earth acting on the bucket and the pulley (gravity) and from the axle acting on the pulley (contact force). If we choose the axis of rotation to coincide with the axle of the pulley, then the axle's force and gravity acting on the pulley each produce no torque. We need only consider the effects of gravity acting on the bucket. 
 Toggle Cloak
idmoda2
 Model: 
 Cloak
idmoda2
 Toggle Cloak
idappa2
 Approach: 
 Cloak
idappa2
 Card
Method 2: Dynamics
Method 2: Dynamics
 Deck of Cards
littledeck
littledeck
 Card
Part A
Part A
 Card
labelPart B
Part B
What is the bucket's speed after falling 5.0 m down the well?
Solution
Again, we will use two methods.
 Deck of Cards
idlittledeck2
 Card
labelMethod 1: Kinematics
Method 1: Kinematics
 Toggle Cloak
idsysb1
 System: 
 Cloak
idsysb1
 Bucket as .
 Toggle Cloak
idintb1
 Interactins: 
 Cloak
idintb1
External influences from the earth (gravity) and the rope (tension).
 Toggle Cloak
idmodb1
 Model: 
 Cloak
idmodb1
One-Dimensional Motion with Constant Acceleration.
 Toggle Cloak
idappb1
 Approach: 
 Cloak
idappb1
Method1
 Card
Method 1: Kinematics
Method 1: Kinematics
Method1
 Card
labelMethod 2: Energy
Method 2: Energy
 Toggle Cloak
idsysb2
 System: 
 Cloak
idsysb2
Treat the bucket, pulley and the earth as a single system.
 Toggle Cloak
idintb2
 Interactions: 
 Cloak
idintb2
Internal interactions of gravity (conservative) and tension from the rope (non-conservative) plus an external influence from the normal force on the pulley (non-conservative).
 Toggle Cloak
idmodb2
 Model: 
 Cloak
idmodb2
.
 Toggle Cloak
idappb2
 Approach: 
 Cloak
idappb2
 Card
Method 2: Energy
Method 2: Energy
 Deck of Cards
littledeck2
littledeck2
 Card
Part B
Part B
 Deck of Cards
bigdeck
bigdeck
 Column
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