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 .

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A string is wrapped over the curved surface of a uniform solid cylinder and the free end is fixed with rigid support. The solid cylinder moves down, unwinding the string. Find the downward acceleration of the solid cylinder.

Important Questions on Rotational Mechanics

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 37

A uniform disk of mass $m$ is released from rest from the rim of a fixed hemispherical bowl so that it rolls along the surface. If the rim of the hemisphere is kept horizontal, find the normal force exerted by the bowl on the disk when it reaches the bottom of the bowl.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 38

There is a rough track, a portion of which is in the form of a cylinder of radius $R$ as shown in the figure. Find the minimum linear speed of a uniform ring of radius $r$ with which it should be set rolling without sliding on the horizontal part so that it can complete round the circle without sliding on the cylindrical part.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 39

A uniform solid sphere of radius $R$ is placed on a smooth horizontal surface. It is pulled by a constant force acting along the tangent from the highest point. Calculate the distance travelled by the centre of mass of the solid sphere during the time it makes one full revolution.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 1>Q 40

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})

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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.

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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.

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$(a)$ 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.

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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 $ω_{0} .$

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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.

A string is wrapped over the curved surface of a uniform solid cylinder and the free end is fixed with rigid support. The solid cylinder moves down, unwinding the string. Find the downward acceleration of the solid cylinder.

Important Questions on Rotational Mechanics

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