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Pulley in rolling

Module by: Sunil Kumar Singh. E-mail the author

Summary: Motion of a pulley of finite mass is analogous to rolling.

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In our earlier treatment on "pulley" in this course, we had limited our consideration to "mass-less" pulley. Here, we shall consider pulley, which has finite mass and is characterized by rolling motion and presence of friction in certain cases.

We need to distinguish two situations (finite mass and mass-less cases) in order to analyze the motion correctly. Mass-less pulley is characterized by the fact that it does not affect the magnitude of tension in the string. It means that tensions on either side in the string remains same.

Analysis of the motion of Pulley in rolling

On the other hand, a pulley of finite mass may rotate, fulfilling the condition of rolling. This means that the length of rope released from the pulley is equal to the distance covered by a point on the rim. If the rolling is accelerated, there may be friction between the pulley surface and the string/ rope, passing over it.

In most cases, however, we may choose to ignore rolling friction as its magnitude is not very appreciable. The pulley, in the figure below, translates in the vertically downward direction, as the string wrapped over it unwinds.

Figure 1: Pulley is rolling on the rope/ string in vertically downward direction.
Pulley
Pulley (pr3.gif)

The length of rope unwound is equal to the vertical distance traveled by the pulley/ disk. Here,

y = θ R y = θ R

This is an analogous situation to the rolling of a disk. As such, equation of rolling and equation of accelerated rolling are valid for the motion of the pulley:

v C = ω R v C = ω R
(1)

a C = α R a C = α R
(2)

In certain situation, the pulley may be fixed to the ceiling as shown in the figure below and hence incapable of translation. We can not say here that pulley is actually rolling. But, the rope translates as much as a point on the rim of the pulley and as such the rope translates at the same velocity and acceleration as that of the center of mass of the pulley, if it were free to translate. We can see that pulley is executing the rotational part of the rolling motion, whereas string, along with attached blocks, is executing the translational part of the rolling motion. Thus, motions of pulley and string together are equivalent to rolling motion.

Figure 2: Fixed pulley and string together executes rolling.
Pulley
Pulley (pr2a.gif)

We must note that friction modifies the tension in the string for the accelerated rolling. For this reason, the tensions in the string may not be same everywhere as shown in the figure above.

We analyze the motion of pulley in same manner as that of a rolling body with the help of two forms of Newton’s second law – one for the linear motion and other for the angular motion. Such consideration of law of motion, however, is conditioned by the equation of rolling and equation of accelerated rolling.

Examples

In this section, we work out with few representative examples :

  • Pulley or disk of finite mass is translating
  • Pulley or disk of finite mass is stationary
  • A mass-less pulley connects rolling motion of a disk to the translation of a block.

Pulley or disk of finite mass is translating

Example 1

Problem : A long rope of negligible mass is wrapped many times over a solid cylinder of mass "m" and radius "R". Other end of the rope is attached to a fixed ceiling and the cylinder is then let go at a given instant. Find the tension and acceleration of the cylinder.

Figure 3: The rolling of the cylinder is guided on the rope.
Rolling down the rope
Rolling down the rope (pr3.gif)

Solution : The rolling of the cylinder is guided on the rope. This is similar to rolling on an incline. We select a coordinate system in which positive y-direction is along the downward motion.

Figure 4: Free body diagram.
Rolling down the rope
Rolling down the rope (pr1.gif)

From the application of Newton's second law for translation in y - direction :

∑ F y = mg - T = m a C ∑ F y = mg - T = m a C
(3)

From the application of Newton's second law for rotation :

τ = T R = I α τ = T R = I α

T = I α R T = I α R
(4)

Note that force due to gravity passes through center of mass and does not constitute a couple to cause angular acceleration. Also, we do not consider sign for angular quantities as we shall be using only the magnitude of angular quantities. Now, from relation of linear and angular accelerations, we have :

a C = α R a C = α R

Putting the value of "α" from above in the equation - 4, we have :

T = I a C R 2 T = I a C R 2

The moment of inertia for solid cylinder about its axis is,

I = m R 2 2 I = m R 2 2

⇒ T = m R 2 a C 2 R 2 ⇒ T = m a C 2 ⇒ T = m R 2 a C 2 R 2 ⇒ T = m a C 2

Substituting the expression of tension in the equation of force analysis in y-direction (equation - 3), we have :

mg - m a C 2 = m a C ⇒ g - a C 2 = a C ⇒ a C ( 1 + 1 2 ) = g ⇒ a C = 2 g 3 mg - m a C 2 = m a C ⇒ g - a C 2 = a C ⇒ a C ( 1 + 1 2 ) = g ⇒ a C = 2 g 3

Putting this value in the expression of "T", we have :

⇒ T = m a C 2 = m g 3 ⇒ T = m a C 2 = m g 3

Pulley or disk of finite mass is stationary

Example 2

Problem : In the “pulley – blocks” arrangement shown in the figure. The masses of the pulley and two blocks are “M”, “ m 1 m 1 ” and “ m 2 m 2 " respectively. If there is no slipping between pulley and rope, then find (i) acceleration of the blocks and (ii) tensions in the string.

Figure 5: There is no slipping between pulley and rope.
Pulley – blocks system
Pulley – blocks system (pr5.gif)

Solution : The pulley has certain mass and rotates to facilitate the accelerations of the blocks in vertical direction. The rotation of pulley of finite mass, therefore, requires certain torque. This means that there is net force on the pulley. This requirement, in turn, dictates that the tensions in the string are different. Let the tensions in the string be “ T 1 T 1 “ and “ T 2 T 2 “. Since the rope is inextensible, the different points on the rope has same acceleration as that of the rolling i.e. a C a C . Let m 2 > m 1 m 2 > m 1 .

Figure 6: There is no slipping between pulley and rope.
Pulley – blocks system
Pulley – blocks system (pr2a.gif)

The magnitude of torque on the pulley (we neglect the consideration of direction) is :

T = ( T 2 - T 1 ) x R T = ( T 2 - T 1 ) x R

The pulley is fixed to the ceiling. As such, it is constrained not to translate in response to the net vertical force on it. Instead, the net vertical force causes translational acceleration of the masses attached to the string. The length of string released from the pulley is equal to the distance covered by a point on the rim of the pulley. It means that the linear acceleration of the string is related to angular acceleration of the pulley by the equation of accelerated rolling. Hence,

a C = α R a C = α R

The directions of angular and linear accelerations are as shown in the figure. Now, considering pulley and blocks, we have three equations as :

(i) Rotation of pulley :

( T 2 - T 1 ) x R = I α = 1 2 x M R 2 α ( T 2 - T 1 ) x R = I α = 1 2 x M R 2 α

⇒ α = 2 ( T 2 - T 1 ) M R ⇒ α = 2 ( T 2 - T 1 ) M R

and

⇒ a C = α R = 2 ( T 2 - T 1 ) M ⇒ a C = α R = 2 ( T 2 - T 1 ) M
(5)

(ii) Translation of m 1 m 1 :

T 1 - m 1 g = m 1 a C T 1 - m 1 g = m 1 a C
(6)

(iii) Translation of m2 :

m 2 g - T 2 = m 2 a C m 2 g - T 2 = m 2 a C
(7)

Putting values of " T 1 T 1 " and " T 2 T 2 " from equations 6 and 7, in the expression of acceleration, we have :

⇒ a C = 2 ( m 2 g - m 2 a C - m 1 g - m 1 a C ) M ⇒ a C = 2 ( m 2 g - m 2 a C - m 1 g - m 1 a C ) M

Reaaranging, ⇒ a C ( M + 2 m 1 + 2 m 2 ) = 2 g ( 2 m 2 - m 1 ) ⇒ a C ( M + 2 m 1 + 2 m 2 ) = 2 g ( 2 m 2 - m 1 )

⇒ a C = 2 g ( 2 m 2 - m 1 ) ( M + 2 m 1 + 2 m 2 ) ⇒ a C = 2 g ( 2 m 2 - m 1 ) ( M + 2 m 1 + 2 m 2 )

Putting values of " a C a C " from equation - 5, we have :

⇒ T 1 = m 1 g + m 1 x 2 g ( 2 m 2 - m 1 ) ( M + 2 m 1 + 2 m 2 ) ⇒ T 1 = m 1 g + m 1 x 2 g ( 2 m 2 - m 1 ) ( M + 2 m 1 + 2 m 2 )

Rearranging, we have :

⇒ T 1 = m 1 g ( M + 4 m 2 ) ( M + 2 m 1 + 2 m 2 ) ⇒ T 1 = m 1 g ( M + 4 m 2 ) ( M + 2 m 1 + 2 m 2 )

Similarly,

⇒ T 2 = m 2 g ( M + 4 m 1 ) ( M + 2 m 1 + 2 m 2 ) ⇒ T 2 = m 2 g ( M + 4 m 1 ) ( M + 2 m 1 + 2 m 2 )

A mass-less pulley connects rolling motion of a disk to the translation of a block

Example 3

Problem : In the “pulley – blocks” arrangement, a string is wound over a circular cylinder of mass “M”. The string passes over a mass-less pulley as shown in the figure. The other end of the string is attached to a block of mass “m”. If the cylinder is rolling on the surface towards right, then find accelerations of the cylinder and block.

Figure 7: Rolling of cylinder and translation of block
Combination of motions via pulley
Combination of motions via pulley (pr6.gif)

Solution : In this question, string passes over a mass-less pulley. It means that the tension in the string through out is same. Since string is attached to the top of the cylinder, the tension force acts tangentially to the cylinder in rolling. In this case, the friction is acting in forward direction.

The linear acceleration of the top position and center of mass are different. As such, linear accelerations of the cylinder and blocks are different. Let the accelerations of cylinder and block be “ a 1 a 1 “and “ a 2 a 2 ” respectively.

Figure 8: Rolling of cylinder and translation of block
Combination of motions via pulley
Combination of motions via pulley (pr7.gif)

(i) Rolling of cylinder :

The magnitude acceleration of the rolling cylinder (i.e. its COM) is :

a C = a 1 = α R a C = a 1 = α R
(8)

Now, we need to relate the of acceleration of the cylinder to that of string (and hence that of block attached to it). Two accelerations are perpendicular to each other and hence in different directions. Thus, we consider only magnitude here. Now, we know that the magnitude of velocity of the top point of the cylinder is related to that of COM as :

v T = 2 v C v T = 2 v C

Differentiating with respect to time, the magnitude of the acceleration of the block is :

a 2 = a T = 2 a C = 2 a 1 = 2 α R a 2 = a T = 2 a C = 2 a 1 = 2 α R
(9)

(ii) Translation of cylinder :

Applying Newton’s second law for translation,

T + f S = M a 1 T + f S = M a 1
(10)

(iii) Rotation of cylinder :

Applying Newton’s second law for rotation,

( T - f S ) x R = I α = 1 2 x M R 2 α ( T - f S ) x R = I α = 1 2 x M R 2 α

⇒ α = 2 ( T - f S ) M R ⇒ α = 2 ( T - f S ) M R

⇒ α M R = 2 T - f S ⇒ α M R = 2 T - f S

For rolling, we use relation of accelerations as given by equation - 9 :

⇒ 2 T - 2 f S = M a 1 ⇒ 2 T - 2 f S = M a 1

Putting values of " f S f S " from equation - 10,

⇒ 2 T - 2 ( M a 1 - T ) = M a 1 ⇒ 2 T - 2 ( M a 1 - T ) = M a 1

⇒ T = 3 M a 1 4 ⇒ T = 3 M a 1 4
(11)

(iv) Translation of block :

The free body diagram of the block is shown in the figure. Here,

Figure 9: The block is accelerating down.
Free body diagram of block
Free body diagram of block (pr8.gif)

m g - T = m a 2 m g - T = m a 2

Substituting value of "T" from equation – 11,

m a 2 - m g = T = 3 M a 1 4 m a 2 - m g = T = 3 M a 1 4

Substituting this value of " a 1 a 1 " in terms of " a 2 a 2 " from equation – 9 and solving for " a 2 a 2 " ,

⇒ a 2 = 8 m g ( 8 m + 3 M ) ⇒ a 2 = 8 m g ( 8 m + 3 M )

As the magnitude of " a 1 a 1 " is given by :

⇒ a 1 = a 2 2 = 4 m g ( 8 m + 3 M ) ⇒ a 1 = a 2 2 = 4 m g ( 8 m + 3 M )

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