A uniform hollow sphere of mass m = 1 kg is placed on a rough horizontal surface for which the coefficient of static friction between the surfaces in contact is μ = 2 5 . Find the maximum constant force which can be applied at the highest point in the horizontal direction so that the sphere can roll without slipping. ( g = 10 m s - 2 )

In absence of friction if force F is applied then linear acceleration a and angular acceleration α will be a = F m and α = τ I = F R 2 3 m R 2 = 3 F 2 m R as α R > a ⇒ In absence of friction there will be backward slipping so, friction force f will act along the same direction as that of applied force F . So, a = F + f m . . i and α = F - f R 2 3 m R 2 = 3 F - f 2 m R … ii As for pure rolling a = α R and using i and ii F + f m = 3 F - f 2 m R R ⇒ 2 F + f = 3 F - f ⇒ F = 5 f As maximum value possible for f = μ m g = 2 5 × 1 kg × 10 m s - 2 = 4 N ⇒ The maximum possible value of F = 5 × 4 N = 20 N

A uniform hollow sphere of mass m = 1 kg is placed on a rough horizontal surface for which the coefficient of static friction between the surfaces in contact is μ = 2 5 . Find the maximum constant force which can be applied at the highest point in the horizontal direction so that the sphere can roll without slipping. ( g = 10 m s - 2 )

In absence of friction if force F is applied then linear acceleration a and angular acceleration α will be a = F m and α = τ I = F R 2 3 m R 2 = 3 F 2 m R as α R > a ⇒ In absence of friction there will be backward slipping so, friction force f will act along the same direction as that of applied force F . So, a = F + f m . . i and α = F - f R 2 3 m R 2 = 3 F - f 2 m R … ii As for pure rolling a = α R and using i and ii F + f m = 3 F - f 2 m R R ⇒ 2 F + f = 3 F - f ⇒ F = 5 f As maximum value possible for f = μ m g = 2 5 × 1 kg × 10 m s - 2 = 4 N ⇒ The maximum possible value of F = 5 × 4 N = 20 N

A uniform solid sphere is placed on a smooth horizontal surface. An impulse I is given horizontally to the sphere at a height h = 4 R / 5 above the centre line. m and R are mass and radius of the sphere, respectively. a Find the angular velocity of the sphere and linear velocity of centre of mass of the sphere after impulse. b Find the minimum time after which the highest point B will touch the ground, c Find the displacement of the centre of mass during this interval.

a Angular impulse about a point is equal to the product of impulse and the perpendicular distance of application of impulse from the point, mathematically J = I r ⊥ = I h … i Now, angular impulse = change in angular momentum = Moment of inertia × change in angular velocity ⇒ J = 2 5 m R 2 ∆ ω ⇒ I h = 2 m R 2 ∆ ω 5 ⇒ I 4 R 5 = 2 m R 2 ∆ ω 5 ⇒ ∆ ω = 2 I m R . As initial angular velocity is zero ⇒ ω = 2 I m R . Again impulse = change in linear momentum = Mass × change in velocity ⇒ I = m ∆ v ⇒ ∆ v = I m . As initial velocity is zero ⇒ v = I m . b The highest point B will touch the ground when the sphere rotates by an angle π ⇒ ω t = π ⇒ t = π ω = π 2 I m R = m R π 2 I . c As COM is moving horizontally with constant velocity, displacement at the same time t will be S = v t = I m m R π 2 I = π R 2 .

HARD

JEE Main/Advance

IMPORTANT

Earn 100

A uniform hollow sphere of mass $m = 1 kg$ is placed on a rough horizontal surface for which the coefficient of static friction between the surfaces in contact is $μ = \frac{2}{5}$ . Find the maximum constant force which can be applied at the highest point in the horizontal direction so that the sphere can roll without slipping. $(g = 10 m s^{- 2})$

Important Questions on Rotational Mechanics

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 41

A uniform rod of length $l$ and mass $4 m$ lies on a frictionless horizontal surface on which it is free to move anyway. A ball of mass $m$ moving with speed $v$ as shown in the figure. It collides with the rod at one of the ends. If the ball comes to rest immediately after collision then find out the angular velocity $ω$ of the rod just after the collision.

Question Image

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 42

A uniform solid sphere is placed on a smooth horizontal surface. An impulse $I$ is given horizontally to the sphere at a height $h = 4 R / 5$ above the centre line. $m$ and $R$ are mass and radius of the sphere, respectively.

Question Image

$(a)$ Find the angular velocity of the sphere and linear velocity of centre of mass of the sphere after impulse.

$(b)$ Find the minimum time after which the highest point $B$ will touch the ground,

$(c)$ Find the displacement of the centre of mass during this interval.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 43

A uniform disc of radius $R = 0.2 m$ kept over a rough horizontal surface is given velocity $v_{0}$ and angular velocity $ω_{0} .$ After some time its kinetic energy becomes zero. If $v_{0} = 10 m s^{- 1},$ find $ω_{0} .$

Question Image

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 44

A solid cubical block of mass

m

and side

a

slides down a rough inclined plane of inclination

θ

with a constant speed. Calculate the torque of the normal force acting on the block about its centre and the perpendicular distance

x

from centre of mass at which it is acting.

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 45

A block hangs from a string wrapped on a disc of radius

20 cm

free to rotate about its axis which is fixed in a horizontal position. If the angular speed of the disc is

10 rad s^{- 1}

at some instant, with what speed is the block going down at that instant?

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 46

A uniform circular disc

A

of radius

r

is made from a copper plate of thickness

t

and another uniform circular disc

B

of radius

2 r

is made from a copper plate of thickness

\frac{t}{2}

. The relation between the moments of inertia

I_{A} and I_{B}

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 47

The moment of inertia of a uniform semicircular wire of mass

M

and radius

r

about a line perpendicular to the plane of the wire through the centre is:

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 48

Let

I_{A} and I_{B}

be the moments of inertia of two solid cylinders of identical geometrical shape and size about their axes, the first made of aluminium and the second of iron.

Important Questions on Rotational Mechanics

Learn and Prepare for any exam you want !