In the previous section, we saw the maximum force that can be applied on a wheel which is not able to self propagate. In this section we will see the case of objects rolling along inclined planes
1. Consider the disc shown in fig.7.147 (a) below
• It has a mass m and radius R
• It is rolling down an inclined plane. The angle of inclination is θ
■ We want the disc to have 'pure rolling'. That is., there must not be any slip
2. This disc is not able to self propagate because there is no engine attached
• But it's self weight will cause it to move
• This will be clear when we draw all the force vectors acting on the wheel. They are shown in fig.b
We have:
(i) The weight $\mathbf\small{m|\vec{g}|}$ acting downwards from the center C
♦ The component $\mathbf\small{m|\vec{g}|\cos \theta}$ acts parallel to the inclined plane
♦ The component $\mathbf\small{m|\vec{g}|\sin \theta}$ acts perpendicular to the inclined plane
(ii) The normal reaction $\mathbf\small{|\vec{N}|}$ acting perpendicular to the plane
♦ This acts at the point of contact
♦ The magnitude is equal to $\mathbf\small{m|\vec{g}|\cos \theta}$
(This force is not shown in the fig)
(iii) The frictional force $\mathbf\small{|\vec{f_s}|=\mu_s|\vec{N}|=\mu_sm|\vec{g}|\cos \theta}$
♦ This force acts parallel to the inclined plane
• Note that, we use the 𝝁s instead of 𝝁k because, we want pure rolling
• Also recall that, when there is no self propagation, the frictional force will be opposite to the direction of motion
3. Our aim is to find the maximum force acting on the disc so that no slipping occurs
• We will separate the calculations into two sets:
Set 1 comprises of the calculations related to rotation
Set 2 comprises of the calculations related to translation
4. Set 1:
(i) If there is a rolling motion, there has to be a torque
• This torque is provided by the frictional force $\mathbf\small{|\vec{f_s}|}$
• Because, all other forces are acting through the center of the wheel
• So we have: $\mathbf\small{|\vec{\tau}|=|\vec{f_s}|R=\mu_sm|\vec{g}|R \cos \theta}$
• $\mathbf\small{\mu_sm|\vec{g}|R \cos \theta}$ is the maximum frictional force possible
• So we can write: $\mathbf\small{|\vec{\tau}_{max}|=\mu_sm|\vec{g}|R \cos \theta}$
(ii) But $\mathbf\small{|\vec{\tau}|=I\,|\vec{\alpha}|}$
• Where:
♦ $\mathbf\small{I}$ is the moment of inertia of the wheel
♦ $\mathbf\small{|\vec{\alpha}|}$ is the angular acceleration of the wheel
(iii) For pure rolling, we have Eq.7.36 that we derived in the previous section: $\mathbf\small{|\vec{a}|=R\,|\vec{\alpha}|}$
• Rearranging this, we get: $\mathbf\small{|\vec{\alpha}|=\frac{|\vec{a}|}{R}}$
(iv) For our present case, we must consider the maximum allowable angular acceleration
• So we must denote it as $\mathbf\small{|\vec{\alpha}_{max}|}$
• So we have: $\mathbf\small{|\vec{\alpha}_{max}|=\frac{|\vec{a}_{max}|}{R}}$
(v) Thus from (ii), we get: $\mathbf\small{|\vec{\tau}_{max}|=I\,\frac{|\vec{a}_{max}|}{R}}$
• We know the I of the disc: $\mathbf\small{I=\frac{mR^2}{2}}$
• So we get: $\mathbf\small{|\vec{\tau}_{max}|=\left(\frac{mR^2}{2}\right)\,\frac{|\vec{a}_{max}|}{R}=\frac{mR|\vec{a}_{max}|}{2}}$
(vi) So the result in (i) becomes: $\mathbf\small{\frac{mR|\vec{a}_{max}|}{2}=\mu_sm|\vec{g}|R \cos \theta}$
• From this we get: $\mathbf\small{|\vec{a}_{max}|= 2\mu_s|\vec{g}|\cos \theta}$
5. Set 2:
We have: Net force = mass × acceleration
• In our present case:
Net force = $\mathbf\small{m|\vec{g}|\sin \theta-|\vec{f_s}|}$
= $\mathbf\small{m|\vec{g}|\sin \theta-\mu_s\,m|\vec{g}|\cos \theta}$
= $\mathbf\small{m|\vec{a}_{max}|}$
6. The two sets are complete
• We take the result from Set 1 (vi)
• And apply it in the result from Set 2
• We get:
Net force
= $\mathbf\small{m|\vec{g}|\sin \theta-\mu_s\,m|\vec{g}|\cos \theta}$
= $\mathbf\small{m\times 2\mu_s|\vec{g}|\cos \theta}$
$\mathbf\small{\Rightarrow m|\vec{g}|\sin \theta=3\mu_s\,m|\vec{g}|\cos \theta}$
7. The quantity $\mathbf\small{|\vec{F}_{max}|}$ is the force that causes the disc to roll
• It is denoted as $\mathbf\small{|\vec{F}_{max}|}$
8. Thus we get:
Eq.7.38:
For a disc rolling down an inclined plane:
$\mathbf\small{|\vec{F}_{max}|}$
= $\mathbf\small{m|\vec{g}|\sin \theta}$
= $\mathbf\small{m\times2\mu_s|\vec{g}|\cos \theta+m\times\mu_s|\vec{g}|\cos \theta=3\mu_sm|\vec{g}|\cos \theta}$
= $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
• Note that, this equation is valid only for a disc and a solid cylinder. For other types of objects, suitable equations must be derived using the appropriate values of I
9. We see that $\mathbf\small{m|\vec{g}|\sin \theta}$ is the force that causes the disc to move
• This $\mathbf\small{m|\vec{g}|\sin \theta}$ must not exceed a 'certain value'
• This 'certain value' is: $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
10. So we can write:
$\mathbf\small{m|\vec{g}|\sin \theta}$ should always be less than or equal to $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
• That is: $\mathbf\small{m|\vec{g}|\sin \theta\;\;\leq \;\;3\mu_sm|\vec{g}|\cos \theta}$
• Where
♦ $\mathbf\small{m|\vec{g}|\sin \theta}$ is the actual force acting
♦ $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ is the maximum allowable force
11. If the left side ever becomes greater than the right side, we will have to make adjustments
• We cannot change mass because, it occurs on both sides
• We see that 𝝁s occurs on the right side only
• So, if the left side increases beyond the limit, we can increase the right side by increasing 𝝁s.
An example:
• m = 5 kg, θ = 35o and 𝝁s = 0.2
Then:
(i) Actual force acting = $\mathbf\small{m|\vec{g}|\sin \theta}$ = 28.1 N
(ii) Maximum allowable force = $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ = 24.1 N
(iii) We have to increase (ii). For that, we will increase the 𝝁s. This can be achieved by increasing the roughness of surface
(iv) But what must be the new value of 𝝁s ?
• To find that, we consider the inequality:
$\mathbf\small{m|\vec{g}|\sin \theta\;\;\leq \;\;3\mu_sm|\vec{g}|\cos \theta}$
• If we multiply or divide both sides of an inequality by a same quantity, the 'inequality' will not change
(v) Here, we will divide both sides by $\mathbf\small{m|\vec{g}|\cos \theta}$. We get:
$\mathbf\small{\tan \theta\;\;\leq \;\;3\mu_s}$
• Dividing both sides by 3, we get:
$\mathbf\small{\frac{\tan \theta}{3}\;\;\leq \;\;\mu_s}$
(vi) That means 𝝁s must be just greater than $\mathbf\small{\frac{\tan \theta}{3}}$
• If this condition is satisfied, the allowable force will be just sufficient
12. So for our present case, 𝝁s must be just greater than (tan 35⁄3) = 0.233
• Let us provide a 𝝁s of 0.25
13. Let us check with the new data:
• m = 5 kg, θ = 35o and 𝝁s = 0.25
Then:
(i) Actual force acting = $\mathbf\small{m|\vec{g}|\sin \theta}$ = 28.1 N
(ii) Maximum allowable force = $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ = 30.1 N
Note: In actual situations, various safety factors must be multiplied to the newly obtained 𝝁s. Only then it will be considered safe. We will learn about them in higher classes
14. If the 𝝁s cannot be changed, then the only option is to change θ
(i) To find the new value of θ, we consider the inequality:
$\mathbf\small{m|\vec{g}|\sin \theta\;\;\leq \;\;3\mu_sm|\vec{g}|\cos \theta}$
• If we multiply or divide both sides of an inequality by a same quantity, the 'inequality' will not change
(ii) Here, we will divide both sides by $\mathbf\small{m|\vec{g}|\cos \theta}$. We get:
$\mathbf\small{\tan \theta\;\;\leq \;\;3\mu_s}$
Thus we get:
7.39: $\mathbf\small{\theta\;\;\leq \;\;\tan^{-1}(3\mu_s)}$
• Note that, this relation is valid only for a disc and a solid cylinder. For other types of objects, suitable relations must be derived using the appropriate values of I
(iii) That means, θ should be brought down so that, it is just less than $\mathbf\small{tan^{-1}(3\mu_s)}$
(iv) If this condition is satisfied, the allowable force will be just sufficient
15. So for our present case, θ must be just less than [tan-1 (3 × 0.2)] = 30.96o
• Let us provide a θ of 30
16. Let us check with the new data:
• m = 5 kg, θ = 30o and 𝝁s = 0.2
Then:
(i) Actual force acting = $\mathbf\small{m|\vec{g}|\sin \theta}$ = 24.5 N
(ii) Maximum allowable force = $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ = 25.5 N
Note: In actual situations, various safety factors must be multiplied to the newly obtained θ. Only then it will be considered safe. We will learn about them in higher classes
1. We have a disc rolling down an inclined plane
• From the given data, we are able to find the following items:
(i) The maximum force that will be causing the motion:
$\mathbf\small{|\vec{F}_{max}|}$ = $\mathbf\small{m|\vec{g}|\sin \theta}$
(ii) The maximum allowable force:
$\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
• Using the above two items, we can determine whether the disc will slip or not
2. The condition at which the two items become equal is the 'limiting condition'
• At that condition, 'maximum allowable net force' will be acting on the disc
• This is given by: Net force = $\mathbf\small{m|\vec{g}|\sin \theta-|\vec{f_s}|}$
= $\mathbf\small{m|\vec{g}|\sin \theta-\mu_s\,m|\vec{g}|\cos \theta}$
3. That means, at that condition, the disc will be experiencing $\mathbf\small{|\vec{a}_{max}|}$ the 'maximum allowable linear acceleration'
• And we know the method to find that maximum allowable linear acceleration:
$\mathbf\small{|\vec{a}_{max}|= 2\mu_s|\vec{g}|\cos \theta}$
This will occur under the following conditions:
• 𝝁s is greater than the limiting value
or
• θ is less than the limiting value
2. In such a situation, the acceleration $\mathbf\small{|\vec{a}|}$ experienced by the disc will be less than $\mathbf\small{|\vec{a}_{max}|}$
3. In such a situation, the force ($\mathbf\small{m|\vec{g}|\sin \theta}$) which causes motion of the disc will be well within the safe limit
4. We want to know the $\mathbf\small{|\vec{a}|}$ in such a situation
5. This is a special situation
The reason for the situation becoming 'special' can be written in 6 steps:
(i) We usually calculate the static frictional force using 𝝁s
(ii) But such a calculation gives the 'maximum static frictional force which is possible'
(iii) When the disc is rolling well with in the safe limit, this maximum static frictional force will not be acting on it
(iv) Because, we know that, static friction is a self adjusting force
♦ It increases when the applied force increases
♦ It decreases when the applied force decreases
■ Here we experience the difference between rolling motion and sliding motion
• The sliding motion begins only when the full available frictional force is exceeded
• The rolling motion does not wait for 'the full available frictional force to be exceeded'
(v) In our present case, the applied force $\mathbf\small{m|\vec{g}|\sin \theta}$ is so less that, the full static frictional force is not mobilized
• So we do not know exactly how much static friction is acting
• So we cannot use 𝝁s.
6. However, we can derive a formula, which does not involve the coefficient
We will separate the calculations into two sets:
• Set 1 comprises of the calculations related to rotation
• Set 2 comprises of the calculations related to translation
7. Set 1:
(i) $\mathbf\small{|\vec{\tau}|=I|\vec{\alpha}|=\frac{mR^2}{2}|\vec{\alpha}|=\frac{mR^2}{2}\times\frac{|\vec{a}|}{R}=\frac{mR|\vec{a}|}{2}}$
(ii) But $\mathbf\small{|\vec{\tau}|=\mu m|\vec{g}|\cos \theta \times R}$
(iii) Equating the results, we get: $\mathbf\small{|\vec{\tau}|=\mu\,m|\vec{g}|\cos \theta \times R=\frac{mR|\vec{a}|}{2}}$
$\mathbf\small{\Rightarrow \mu|\vec{g}|\cos \theta=\frac{|\vec{a}|}{2}}$
8. Set 2:
Net force = $\mathbf\small{m|\vec{g}|\sin \theta-\mu\;m|\vec{g}|\cos \theta=m|\vec{a}|}$
9. The two sets are complete
• Now we take the result from Set 1 (iii)
• And apply it in the result from Set 2
We get:
Net force = $\mathbf\small{m|\vec{g}|\sin \theta-m\times\frac{|\vec{a}|}{2}=m|\vec{a}|}$
$\mathbf\small{\Rightarrow m|\vec{g}|\sin \theta=\frac{3m|\vec{a}|}{2}}$
$\mathbf\small{\Rightarrow |\vec{a}|=\frac{2|\vec{g}|\sin \theta}{3}}$
10. We can write:
When a disc is rolling down an incline with out slipping, the linear acceleration is given by:
Eq.7.40: $\mathbf\small{|\vec{a}|=\frac{2|\vec{g}|\sin \theta}{3}}$
• Note that, this equation is valid only for a disc and a solid cylinder. For other types of objects, suitable equations must be derived using the appropriate values of I
Solved example 7.51
A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30o. The coefficient of static friction 𝝁s = 0.25.
(a) How much is the force of friction acting on the cylinder ?
(b) What is the work done against friction during rolling ?
(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid,
and not roll perfectly ?
Solution:
• Both solid cylinder and disc have the same I. So the equations that we derived above are applicable for this problem
• First we need to know whether the maximum frictional force is mobilized or not
• For that we do the following 4 steps:
1. We will use the relation in Eq.7.38: For a disc rolling down an inclined plane:
$\mathbf\small{|\vec{F}_{max}|}$ = $\mathbf\small{m|\vec{g}|\sin \theta}$
• This maximum force must be less than or equal to $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
2. Substituting the values we get:
$\mathbf\small{m|\vec{g}|\sin \theta=49\,\rm{N}}$
3. Similarly, we get: $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta=63.65\,\rm{N}}$
4. Result in (2) is less than the result in (3)
• So it is safe. The cylinder will not slip. Also there is no need to mobilize the maximum available frictional force
• Now we can find the answers to the given questions
Part (a):
1. To find the actual acceleration, we use
Eq.7.40: $\mathbf\small{|\vec{a}|=\frac{2|\vec{g}|\sin \theta}{3}}$
• Substituting the values, we get: $\mathbf\small{|\vec{a}|=3.27\,\rm{m\,s^{-2}}}$
2. We have:
Net force = Force acting downwards along the slope - Frictional force
⇒ Net force = [$\mathbf\small{m|\vec{g}|\sin \theta}$ - Frictional force]
⇒ Net force = [49 - Frictional force]
3. But net force = Mass × acceleration = (10 × 3.27) = 32.7 N
4. So the result in (2) becomes:
32.7 = [49 - Frictional force]
⇒ Frictional force = 49 - 32.7 = 16.3 N
Part (b):
1. We have: Work done = Force × Displacement
2. The frictional force acts on every particle on the periphery of the cylinder
3. Consider the instant at which a particle experiences the frictional force
• At that instant, the displacement of that particle is zero
• At the next instant, the frictional force no longer acts on that particle
♦ Because, the succeeding particle takes up it's position
4. So displacement is always zero
5. So no work is done against the frictional force
Part (c):
• We will use the relation in 7.39: $\mathbf\small{\theta\;\;\leq \;\;\tan^{-1}(3\mu_s)}$
• Substituting the values we get: θ ≤ tan-1 (3 × 0.25)
⇒ θ ≤ tan-1 (0.75)
⇒ θ ≤ 36.87
• So if θ is greater than 36.87o, the cylinder will begin to skid
1. Consider the disc shown in fig.7.147 (a) below
 |
| Fig.7.147(a) |
• It is rolling down an inclined plane. The angle of inclination is θ
■ We want the disc to have 'pure rolling'. That is., there must not be any slip
2. This disc is not able to self propagate because there is no engine attached
• But it's self weight will cause it to move
• This will be clear when we draw all the force vectors acting on the wheel. They are shown in fig.b
We have:
(i) The weight $\mathbf\small{m|\vec{g}|}$ acting downwards from the center C
♦ The component $\mathbf\small{m|\vec{g}|\cos \theta}$ acts parallel to the inclined plane
♦ The component $\mathbf\small{m|\vec{g}|\sin \theta}$ acts perpendicular to the inclined plane
(ii) The normal reaction $\mathbf\small{|\vec{N}|}$ acting perpendicular to the plane
♦ This acts at the point of contact
♦ The magnitude is equal to $\mathbf\small{m|\vec{g}|\cos \theta}$
(This force is not shown in the fig)
(iii) The frictional force $\mathbf\small{|\vec{f_s}|=\mu_s|\vec{N}|=\mu_sm|\vec{g}|\cos \theta}$
♦ This force acts parallel to the inclined plane
• Note that, we use the 𝝁s instead of 𝝁k because, we want pure rolling
• Also recall that, when there is no self propagation, the frictional force will be opposite to the direction of motion
3. Our aim is to find the maximum force acting on the disc so that no slipping occurs
• We will separate the calculations into two sets:
Set 1 comprises of the calculations related to rotation
Set 2 comprises of the calculations related to translation
4. Set 1:
(i) If there is a rolling motion, there has to be a torque
• This torque is provided by the frictional force $\mathbf\small{|\vec{f_s}|}$
• Because, all other forces are acting through the center of the wheel
• So we have: $\mathbf\small{|\vec{\tau}|=|\vec{f_s}|R=\mu_sm|\vec{g}|R \cos \theta}$
• $\mathbf\small{\mu_sm|\vec{g}|R \cos \theta}$ is the maximum frictional force possible
• So we can write: $\mathbf\small{|\vec{\tau}_{max}|=\mu_sm|\vec{g}|R \cos \theta}$
(ii) But $\mathbf\small{|\vec{\tau}|=I\,|\vec{\alpha}|}$
• Where:
♦ $\mathbf\small{I}$ is the moment of inertia of the wheel
♦ $\mathbf\small{|\vec{\alpha}|}$ is the angular acceleration of the wheel
(iii) For pure rolling, we have Eq.7.36 that we derived in the previous section: $\mathbf\small{|\vec{a}|=R\,|\vec{\alpha}|}$
• Rearranging this, we get: $\mathbf\small{|\vec{\alpha}|=\frac{|\vec{a}|}{R}}$
(iv) For our present case, we must consider the maximum allowable angular acceleration
• So we must denote it as $\mathbf\small{|\vec{\alpha}_{max}|}$
• So we have: $\mathbf\small{|\vec{\alpha}_{max}|=\frac{|\vec{a}_{max}|}{R}}$
(v) Thus from (ii), we get: $\mathbf\small{|\vec{\tau}_{max}|=I\,\frac{|\vec{a}_{max}|}{R}}$
• We know the I of the disc: $\mathbf\small{I=\frac{mR^2}{2}}$
• So we get: $\mathbf\small{|\vec{\tau}_{max}|=\left(\frac{mR^2}{2}\right)\,\frac{|\vec{a}_{max}|}{R}=\frac{mR|\vec{a}_{max}|}{2}}$
(vi) So the result in (i) becomes: $\mathbf\small{\frac{mR|\vec{a}_{max}|}{2}=\mu_sm|\vec{g}|R \cos \theta}$
• From this we get: $\mathbf\small{|\vec{a}_{max}|= 2\mu_s|\vec{g}|\cos \theta}$
5. Set 2:
We have: Net force = mass × acceleration
• In our present case:
Net force = $\mathbf\small{m|\vec{g}|\sin \theta-|\vec{f_s}|}$
= $\mathbf\small{m|\vec{g}|\sin \theta-\mu_s\,m|\vec{g}|\cos \theta}$
= $\mathbf\small{m|\vec{a}_{max}|}$
6. The two sets are complete
• We take the result from Set 1 (vi)
• And apply it in the result from Set 2
• We get:
Net force
= $\mathbf\small{m|\vec{g}|\sin \theta-\mu_s\,m|\vec{g}|\cos \theta}$
= $\mathbf\small{m\times 2\mu_s|\vec{g}|\cos \theta}$
$\mathbf\small{\Rightarrow m|\vec{g}|\sin \theta=3\mu_s\,m|\vec{g}|\cos \theta}$
7. The quantity $\mathbf\small{|\vec{F}_{max}|}$ is the force that causes the disc to roll
• It is denoted as $\mathbf\small{|\vec{F}_{max}|}$
8. Thus we get:
Eq.7.38:
For a disc rolling down an inclined plane:
$\mathbf\small{|\vec{F}_{max}|}$
= $\mathbf\small{m|\vec{g}|\sin \theta}$
= $\mathbf\small{m\times2\mu_s|\vec{g}|\cos \theta+m\times\mu_s|\vec{g}|\cos \theta=3\mu_sm|\vec{g}|\cos \theta}$
= $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
• Note that, this equation is valid only for a disc and a solid cylinder. For other types of objects, suitable equations must be derived using the appropriate values of I
9. We see that $\mathbf\small{m|\vec{g}|\sin \theta}$ is the force that causes the disc to move
• This $\mathbf\small{m|\vec{g}|\sin \theta}$ must not exceed a 'certain value'
• This 'certain value' is: $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
10. So we can write:
$\mathbf\small{m|\vec{g}|\sin \theta}$ should always be less than or equal to $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
• That is: $\mathbf\small{m|\vec{g}|\sin \theta\;\;\leq \;\;3\mu_sm|\vec{g}|\cos \theta}$
• Where
♦ $\mathbf\small{m|\vec{g}|\sin \theta}$ is the actual force acting
♦ $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ is the maximum allowable force
11. If the left side ever becomes greater than the right side, we will have to make adjustments
• We cannot change mass because, it occurs on both sides
• We see that 𝝁s occurs on the right side only
• So, if the left side increases beyond the limit, we can increase the right side by increasing 𝝁s.
An example:
• m = 5 kg, θ = 35o and 𝝁s = 0.2
Then:
(i) Actual force acting = $\mathbf\small{m|\vec{g}|\sin \theta}$ = 28.1 N
(ii) Maximum allowable force = $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ = 24.1 N
(iii) We have to increase (ii). For that, we will increase the 𝝁s. This can be achieved by increasing the roughness of surface
(iv) But what must be the new value of 𝝁s ?
• To find that, we consider the inequality:
$\mathbf\small{m|\vec{g}|\sin \theta\;\;\leq \;\;3\mu_sm|\vec{g}|\cos \theta}$
• If we multiply or divide both sides of an inequality by a same quantity, the 'inequality' will not change
(v) Here, we will divide both sides by $\mathbf\small{m|\vec{g}|\cos \theta}$. We get:
$\mathbf\small{\tan \theta\;\;\leq \;\;3\mu_s}$
• Dividing both sides by 3, we get:
$\mathbf\small{\frac{\tan \theta}{3}\;\;\leq \;\;\mu_s}$
(vi) That means 𝝁s must be just greater than $\mathbf\small{\frac{\tan \theta}{3}}$
• If this condition is satisfied, the allowable force will be just sufficient
12. So for our present case, 𝝁s must be just greater than (tan 35⁄3) = 0.233
• Let us provide a 𝝁s of 0.25
13. Let us check with the new data:
• m = 5 kg, θ = 35o and 𝝁s = 0.25
Then:
(i) Actual force acting = $\mathbf\small{m|\vec{g}|\sin \theta}$ = 28.1 N
(ii) Maximum allowable force = $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ = 30.1 N
Note: In actual situations, various safety factors must be multiplied to the newly obtained 𝝁s. Only then it will be considered safe. We will learn about them in higher classes
14. If the 𝝁s cannot be changed, then the only option is to change θ
(i) To find the new value of θ, we consider the inequality:
$\mathbf\small{m|\vec{g}|\sin \theta\;\;\leq \;\;3\mu_sm|\vec{g}|\cos \theta}$
• If we multiply or divide both sides of an inequality by a same quantity, the 'inequality' will not change
(ii) Here, we will divide both sides by $\mathbf\small{m|\vec{g}|\cos \theta}$. We get:
$\mathbf\small{\tan \theta\;\;\leq \;\;3\mu_s}$
Thus we get:
7.39: $\mathbf\small{\theta\;\;\leq \;\;\tan^{-1}(3\mu_s)}$
• Note that, this relation is valid only for a disc and a solid cylinder. For other types of objects, suitable relations must be derived using the appropriate values of I
(iii) That means, θ should be brought down so that, it is just less than $\mathbf\small{tan^{-1}(3\mu_s)}$
(iv) If this condition is satisfied, the allowable force will be just sufficient
15. So for our present case, θ must be just less than [tan-1 (3 × 0.2)] = 30.96o
• Let us provide a θ of 30
16. Let us check with the new data:
• m = 5 kg, θ = 30o and 𝝁s = 0.2
Then:
(i) Actual force acting = $\mathbf\small{m|\vec{g}|\sin \theta}$ = 24.5 N
(ii) Maximum allowable force = $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$ = 25.5 N
Note: In actual situations, various safety factors must be multiplied to the newly obtained θ. Only then it will be considered safe. We will learn about them in higher classes
Let us write a summary of the above discussion. The summary can be written in 3 steps:
• From the given data, we are able to find the following items:
(i) The maximum force that will be causing the motion:
$\mathbf\small{|\vec{F}_{max}|}$ = $\mathbf\small{m|\vec{g}|\sin \theta}$
(ii) The maximum allowable force:
$\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
• Using the above two items, we can determine whether the disc will slip or not
2. The condition at which the two items become equal is the 'limiting condition'
• At that condition, 'maximum allowable net force' will be acting on the disc
• This is given by: Net force = $\mathbf\small{m|\vec{g}|\sin \theta-|\vec{f_s}|}$
= $\mathbf\small{m|\vec{g}|\sin \theta-\mu_s\,m|\vec{g}|\cos \theta}$
3. That means, at that condition, the disc will be experiencing $\mathbf\small{|\vec{a}_{max}|}$ the 'maximum allowable linear acceleration'
• And we know the method to find that maximum allowable linear acceleration:
$\mathbf\small{|\vec{a}_{max}|= 2\mu_s|\vec{g}|\cos \theta}$
1. But what if the disc is rolling well within the limit?
• 𝝁s is greater than the limiting value
or
• θ is less than the limiting value
2. In such a situation, the acceleration $\mathbf\small{|\vec{a}|}$ experienced by the disc will be less than $\mathbf\small{|\vec{a}_{max}|}$
3. In such a situation, the force ($\mathbf\small{m|\vec{g}|\sin \theta}$) which causes motion of the disc will be well within the safe limit
4. We want to know the $\mathbf\small{|\vec{a}|}$ in such a situation
5. This is a special situation
The reason for the situation becoming 'special' can be written in 6 steps:
(i) We usually calculate the static frictional force using 𝝁s
(ii) But such a calculation gives the 'maximum static frictional force which is possible'
(iii) When the disc is rolling well with in the safe limit, this maximum static frictional force will not be acting on it
(iv) Because, we know that, static friction is a self adjusting force
♦ It increases when the applied force increases
♦ It decreases when the applied force decreases
■ Here we experience the difference between rolling motion and sliding motion
• The sliding motion begins only when the full available frictional force is exceeded
• The rolling motion does not wait for 'the full available frictional force to be exceeded'
(v) In our present case, the applied force $\mathbf\small{m|\vec{g}|\sin \theta}$ is so less that, the full static frictional force is not mobilized
• So we do not know exactly how much static friction is acting
• So we cannot use 𝝁s.
6. However, we can derive a formula, which does not involve the coefficient
We will separate the calculations into two sets:
• Set 1 comprises of the calculations related to rotation
• Set 2 comprises of the calculations related to translation
7. Set 1:
(i) $\mathbf\small{|\vec{\tau}|=I|\vec{\alpha}|=\frac{mR^2}{2}|\vec{\alpha}|=\frac{mR^2}{2}\times\frac{|\vec{a}|}{R}=\frac{mR|\vec{a}|}{2}}$
(ii) But $\mathbf\small{|\vec{\tau}|=\mu m|\vec{g}|\cos \theta \times R}$
(iii) Equating the results, we get: $\mathbf\small{|\vec{\tau}|=\mu\,m|\vec{g}|\cos \theta \times R=\frac{mR|\vec{a}|}{2}}$
$\mathbf\small{\Rightarrow \mu|\vec{g}|\cos \theta=\frac{|\vec{a}|}{2}}$
8. Set 2:
Net force = $\mathbf\small{m|\vec{g}|\sin \theta-\mu\;m|\vec{g}|\cos \theta=m|\vec{a}|}$
9. The two sets are complete
• Now we take the result from Set 1 (iii)
• And apply it in the result from Set 2
We get:
Net force = $\mathbf\small{m|\vec{g}|\sin \theta-m\times\frac{|\vec{a}|}{2}=m|\vec{a}|}$
$\mathbf\small{\Rightarrow m|\vec{g}|\sin \theta=\frac{3m|\vec{a}|}{2}}$
$\mathbf\small{\Rightarrow |\vec{a}|=\frac{2|\vec{g}|\sin \theta}{3}}$
10. We can write:
When a disc is rolling down an incline with out slipping, the linear acceleration is given by:
Eq.7.40: $\mathbf\small{|\vec{a}|=\frac{2|\vec{g}|\sin \theta}{3}}$
• Note that, this equation is valid only for a disc and a solid cylinder. For other types of objects, suitable equations must be derived using the appropriate values of I
Solved example 7.51
A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30o. The coefficient of static friction 𝝁s = 0.25.
(a) How much is the force of friction acting on the cylinder ?
(b) What is the work done against friction during rolling ?
(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid,
and not roll perfectly ?
Solution:
• Both solid cylinder and disc have the same I. So the equations that we derived above are applicable for this problem
• First we need to know whether the maximum frictional force is mobilized or not
• For that we do the following 4 steps:
1. We will use the relation in Eq.7.38: For a disc rolling down an inclined plane:
$\mathbf\small{|\vec{F}_{max}|}$ = $\mathbf\small{m|\vec{g}|\sin \theta}$
• This maximum force must be less than or equal to $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta}$
2. Substituting the values we get:
$\mathbf\small{m|\vec{g}|\sin \theta=49\,\rm{N}}$
3. Similarly, we get: $\mathbf\small{3\mu_sm|\vec{g}|\cos \theta=63.65\,\rm{N}}$
4. Result in (2) is less than the result in (3)
• So it is safe. The cylinder will not slip. Also there is no need to mobilize the maximum available frictional force
• Now we can find the answers to the given questions
Part (a):
1. To find the actual acceleration, we use
Eq.7.40: $\mathbf\small{|\vec{a}|=\frac{2|\vec{g}|\sin \theta}{3}}$
• Substituting the values, we get: $\mathbf\small{|\vec{a}|=3.27\,\rm{m\,s^{-2}}}$
2. We have:
Net force = Force acting downwards along the slope - Frictional force
⇒ Net force = [$\mathbf\small{m|\vec{g}|\sin \theta}$ - Frictional force]
⇒ Net force = [49 - Frictional force]
3. But net force = Mass × acceleration = (10 × 3.27) = 32.7 N
4. So the result in (2) becomes:
32.7 = [49 - Frictional force]
⇒ Frictional force = 49 - 32.7 = 16.3 N
Part (b):
1. We have: Work done = Force × Displacement
2. The frictional force acts on every particle on the periphery of the cylinder
3. Consider the instant at which a particle experiences the frictional force
• At that instant, the displacement of that particle is zero
• At the next instant, the frictional force no longer acts on that particle
♦ Because, the succeeding particle takes up it's position
4. So displacement is always zero
5. So no work is done against the frictional force
Part (c):
• We will use the relation in 7.39: $\mathbf\small{\theta\;\;\leq \;\;\tan^{-1}(3\mu_s)}$
• Substituting the values we get: θ ≤ tan-1 (3 × 0.25)
⇒ θ ≤ tan-1 (0.75)
⇒ θ ≤ 36.87
• So if θ is greater than 36.87o, the cylinder will begin to skid
We have completed a discussion on rolling motion. In the next section, we will see a few more solved examples
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