Answer
Verified
439.2k+ views
Hint: We will first find out the moment of inertia of both the cylinders and then will use the formula of acceleration of an object rolling down an inclined plane without slipping. Then by using the equations of motions, we can find the time for both the cylinders.
Formula used: $a=\dfrac{gsin\theta}{1+\dfrac{I}{MR^2}}$
Complete step by step answer:
Let us assume the mass of both solid and hollow cylinder by M and their radius be R, both rolling down an inclined plane of length $l$ making angle $\theta$ with the horizontal without slipping.
We know that, acceleration of a purely rolling object along an inclined plane without slipping is given by, $a=\dfrac{gsin\theta}{1+\dfrac{I}{MR^2}}$ ………. (i)
where, $\theta$ is the inclination of the plane, I is the moment of inertia of the object about an axis perpendicular to its cross-section, M is the mass of the object and R is its equivalent radius.
Now, as per the question both cylinders were released from rest. So, using the distance formula from the equations of motion, we get $S=\dfrac{1}{2}at^2$
As the length of inclined plane is $l$ and putting the value of acceleration from equation (i), we get $S=\dfrac{1}{2}\times \left(\dfrac{gsin\theta}{1+\dfrac{I}{MR^2}}\right)t^2$
$\implies t=\sqrt{2l\left(\dfrac{1+\dfrac{I}{MR^2}}{gsin\theta}\right)}$ ………. (ii)
For solid cylinder,
Moment of inertia, ${I}_{s} = \dfrac{MR^2}{2}$
Therefore, time taken by the solid cylinder to reach the bottom of incline using equation (ii),
${t}_{s}=\sqrt{2l\left(\dfrac{1+\dfrac{MR^2 /2}{MR^2}}{gsin\theta}\right)}=\sqrt{\dfrac{3l}{gsin\theta}}$ ………. (iii)
Now, for hollow cylinder,
Moment of inertia, ${I}_{h} = MR^2$
Therefore, time taken by the solid cylinder to reach the bottom of incline using equation (ii),
${t}_{h} =\sqrt{2l\left(\dfrac{1+\dfrac{MR^2}{MR^2}}{gsin\theta}\right)}=\sqrt{\dfrac{4l}{gsin\theta}}$ ………. (iv)
So, clearly from the equations (iii) and (iv), we can say that ${t}_{s}<{t}_{h}$.
Thus, the solid cylinder will reach the bottom first. Hence, the correct answer is option D.
Note:
The formula for time that we derived can also be used directly. But for concept clearance, we derived here the formula. We can also use the conservation of energy at the point of release and at the bottom to find the final velocity of the cylinders, which in turn can give the time required.
Formula used: $a=\dfrac{gsin\theta}{1+\dfrac{I}{MR^2}}$
Complete step by step answer:
Let us assume the mass of both solid and hollow cylinder by M and their radius be R, both rolling down an inclined plane of length $l$ making angle $\theta$ with the horizontal without slipping.
We know that, acceleration of a purely rolling object along an inclined plane without slipping is given by, $a=\dfrac{gsin\theta}{1+\dfrac{I}{MR^2}}$ ………. (i)
where, $\theta$ is the inclination of the plane, I is the moment of inertia of the object about an axis perpendicular to its cross-section, M is the mass of the object and R is its equivalent radius.
Now, as per the question both cylinders were released from rest. So, using the distance formula from the equations of motion, we get $S=\dfrac{1}{2}at^2$
As the length of inclined plane is $l$ and putting the value of acceleration from equation (i), we get $S=\dfrac{1}{2}\times \left(\dfrac{gsin\theta}{1+\dfrac{I}{MR^2}}\right)t^2$
$\implies t=\sqrt{2l\left(\dfrac{1+\dfrac{I}{MR^2}}{gsin\theta}\right)}$ ………. (ii)
For solid cylinder,
Moment of inertia, ${I}_{s} = \dfrac{MR^2}{2}$
Therefore, time taken by the solid cylinder to reach the bottom of incline using equation (ii),
${t}_{s}=\sqrt{2l\left(\dfrac{1+\dfrac{MR^2 /2}{MR^2}}{gsin\theta}\right)}=\sqrt{\dfrac{3l}{gsin\theta}}$ ………. (iii)
Now, for hollow cylinder,
Moment of inertia, ${I}_{h} = MR^2$
Therefore, time taken by the solid cylinder to reach the bottom of incline using equation (ii),
${t}_{h} =\sqrt{2l\left(\dfrac{1+\dfrac{MR^2}{MR^2}}{gsin\theta}\right)}=\sqrt{\dfrac{4l}{gsin\theta}}$ ………. (iv)
So, clearly from the equations (iii) and (iv), we can say that ${t}_{s}<{t}_{h}$.
Thus, the solid cylinder will reach the bottom first. Hence, the correct answer is option D.
Note:
The formula for time that we derived can also be used directly. But for concept clearance, we derived here the formula. We can also use the conservation of energy at the point of release and at the bottom to find the final velocity of the cylinders, which in turn can give the time required.
Recently Updated Pages
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Advantages and disadvantages of science
10 examples of friction in our daily life
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference Between Plant Cell and Animal Cell
10 examples of evaporation in daily life with explanations
Give 10 examples for herbs , shrubs , climbers , creepers
Write a letter to the principal requesting him to grant class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE