# Connexions

You are here: Home » Content » Physics for K-12 » Rolling along an incline (check your understanding)

• Why yet another course in physics?
• What is physics?

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

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

# Rolling along an incline (check your understanding)

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

Summary: Objective questions, contained in this module with hidden solutions, help improve understanding of the topics covered under the module "Rolling along an incline".

The questions have been selected to enhance understanding of the topics covered in the module titled " Rolling along an incline ". All questions are multiple choice questions with one or more correct answers. There are two sets of the questions. The “understanding level” questions seek to unravel the fundamental concepts involved, whereas “application level” are relatively difficult, which may interlink concepts from other topics.

Each of the questions is provided with solution. However, it is recommended that solutions may be seen only when your answers do not match with the ones given at the end of this module.

## Understanding level (Rolling along an incline)

### Exercise 1

A solid sphere of mass “m” and radius “r” rolls down an incline of angle “θ” without sliding. What is its acceleration down the incline?

(a) g sin θ (b) 2 g sin θ 3 (c) 2 g sin θ 5 (d) 5 g sin θ 7 (a) g sin θ (b) 2 g sin θ 3 (c) 2 g sin θ 5 (d) 5 g sin θ 7

#### Solution

For the analysis of rolling motion, we have three equations :

(i) Newton’s second law for translation

m g sin θ - f S = m a m g sin θ - f S = m a

(ii) Newton’s second law for rotation

τ = f S r = I α f S r = 2 m r 2 α 5 f S = 2 m r α 5 τ = f S r = I α f S r = 2 m r 2 α 5 f S = 2 m r α 5

(iii) Equation of accelerated rolling

a C = a = α r a C = a = α r

Substituting value of “α” in the equation of rotation,

f S = 2 m a 5 f S = 2 m a 5

Putting this value in the equation of translation,

m g sin θ - 2 m a 5 = m a m g sin θ - 2 m a 5 = m a

a = 5 g sin θ 7 a = 5 g sin θ 7

Hence, option (d) is correct.

### Exercise 2

A solid sphere of mass “m” and radius “r” is rolled up on an incline as shown in the figure. Then,

(a) Friction acts upwards (b) Friction acts downwards (c) Friction accelerates translation (d) Friction accelerates rotation. (a) Friction acts upwards (b) Friction acts downwards (c) Friction accelerates translation (d) Friction accelerates rotation.

#### Solution

The solid sphere rotates clockwise to move up along the incline. The component of gravity along the incline acting downward decelerates translation. The friction, therefore, should appear in such a fashion that (i) there is linear acceleration to counter the deceleration due to gravity and (ii) there is angular deceleration so that equation of accelerated rolling is held.

This means that friction acts upward. It accelerates translation in the direction of motion.

Hence, options (a) and (c) are correct.

### Exercise 3

A block of mass “m” slides down a smooth incline and a sphere of the same mass rolls down another identical incline. If both start their motion simultaneously from the same height on the incline, then

(a) block reaches the bottom first (b) sphere reaches the bottom first (c) both reaches at the same time (d) either of them can reach first, depending on the friction between sphere and incline (a) block reaches the bottom first (b) sphere reaches the bottom first (c) both reaches at the same time (d) either of them can reach first, depending on the friction between sphere and incline

#### Solution

The time taken to reach bottom depends on the translational velocity. In turn, velocity depends on the acceleration of the bodies during motion.

In the first case, the block slides on the smooth plane. Hence, there is no friction between surfaces. The force along the incline is component of force due to gravity in the direction of motion. The translational acceleration of the block is :

a B = g sin θ a B = g sin θ

In the second case, static friction less than maximum static friction operates at the contact point. The net force on the sphere, rolling down the incline, is :

F net = m g sin θ - f S F net = m g sin θ - f S

The corresponding translational acceleration of the sphere is :

a S = g sin θ - f S m a S = g sin θ - f S m

Evidently,

a B > a S a B > a S

Therefore, the block reaches the bottom first.

Hence, option (a) is correct.

## Application level (Rolling along an incline)

### Exercise 4

A solid sphere of mass “m” and radius “r”, rolls down an incline of angle “θ” without sliding. What should be the minimum coefficient of friction (μ) so that the solid sphere rolls down without sliding?

(a) μ > 2 g tan θ 7 (b) μ < 2 g sin θ 5 (c) μ < 2 g cos θ 5 (d) μ > 2 g tan θ 3 (a) μ > 2 g tan θ 7 (b) μ < 2 g sin θ 5 (c) μ < 2 g cos θ 5 (d) μ > 2 g tan θ 3

#### Solution

Rolling requires a minimum threshold value of friction. If the friction is less than this threshold value, then the solid sphere will slide down instead of rolling down. Now, for the analysis of rolling motion, we have three equations :

(i) Newton’s second law for translation

m g sin θ - f S = m a m g sin θ - f S = m a

(ii) Newton’s second law for rotation

τ = f S r = I α f S r = 2 m r 2 α 5 f S = 2 m r α 5 τ = f S r = I α f S r = 2 m r 2 α 5 f S = 2 m r α 5

(iii) Equation of accelerated rolling

a C = a = α r a C = a = α r

Substituting value of “α” in the equation of rotation,

f S = 2 m a 5 f S = 2 m a 5

Putting this value in the equation of translation,

m g sin θ - 2 m a 5 = m a m g sin θ - 2 m a 5 = m a

a = 5 g sin θ 7 a = 5 g sin θ 7

Thus,

f S = 2 m a 5 = 2 m g sin θ 7 f S = 2 m a 5 = 2 m g sin θ 7

The limiting friction corresponding to a given coefficient of friction is :

F S = μ N = μ m g cos θ F S = μ N = μ m g cos θ

For rolling to take place, the coefficient of friction be such that limiting friction corresponding to it is greater than the static friction required to maintain rolling,

μ m g cos θ > 2 m g sin θ 7 μ > 2 g tan θ 7 μ m g cos θ > 2 m g sin θ 7 μ > 2 g tan θ 7

Hence, option (a) is correct.

### Exercise 5

A solid sphere of mass “M” and radius “R” is set to roll down a smooth incline as shown in the figure. If the smooth incline plane is accelerated to the right with acceleration “a”, then what should be the horizontal acceleration of incline such that solid sphere rolls without sliding?

(a) g sin θ (b) g cos θ (c) g tan θ (d) g cot θ (a) g sin θ (b) g cos θ (c) g tan θ (d) g cot θ

#### Solution

There is no friction between smooth incline and the solid sphere. The rolling sphere will roll on smooth plane at constant velocity. Acceleration will tend to slide the solid sphere on a smooth plane, where there is no friction. Hence, net force on the sphere along the incline should be zero so that the sphere keeps rolling with constant velocity.

Now, the incline itself is an accelerated frame of reference. For force analysis, we consider a pseudo force to covert the accelerated reference system to an equivalent inertia reference system. The pseudo force acts through the center of mass and its magnitude is :

F Pseudo = m a F Pseudo = m a

Analyzing force for translation along the incline, we have :

m g sin θ - m a cos θ = 0 m g sin θ - m a cos θ = 0

a = g tan θ a = g tan θ

Hence, option (c) is correct.


1. (d)   2. (a) and (c)    3. (a)
4. (a)   5. (c)


## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks