A force F → = A i ^ + B j ^ is applied to a point whose radius vector, relative to the origin of coordinates O , is equal to r → = a i ^ + b j ^ , where a , b , A and B are constants, and i ^ and j ^ are the unit vectors along x and y -axes. Find the moment N and the arm l of the force F , relative to the point O .

We know that the moment (torque produced due to a force) is given by, ⇒ N → = r → × F → , where r → is the radius vector and F → is the force vector. ⇒ N → = a i ^ + b j ^ × A i ^ + B j ^ ⇒ N → = a B k ^ + A b - k ^ ∴ N → = a B - b A k ^ . As the magnitude of the moment is given by N = F l . So, the arm (lever arm) l of the force will be, l = N F ∴ l = a B - b A A 2 + B 2 .

Consider a bicycle in vertical position accelerating forward without slipping on a straight horizontal road. The combined mass of the bicycle and the rider is M and the magnitude of the accelerating torque applied on the rear wheel by the pedal and gear system is τ . The radius and the moment of inertia of each wheel is R and I (with respect to the axis) respectively. The acceleration due to gravity is g . (a) Draw the free diagram of the system (bicycle and rider ). (b) Obtains the acceleration a in terms of the above-mentioned quantities. a = (c) For simplicity assume that the centre of mass of the system is at height R from the ground and equidistant at 2 R from the centre of each of the wheels. Let μ be the coefficient of friction (both static and dynamic) between the wheels and the ground. Consider M > > I R 2 and no slipping. Obtain the conditions for the maximum acceleration a m of the bike. a m = (d) For μ = 1 . 0 calculate a m . a m =

(a) The friction forces f 1 and f 2 on the wheels and normal N 1 and N 2 on the wheels are shown on figure below, Here, for pure rolling a = α R (b) From the free body diagram, we have, N 1 + N 2 = M g . . . 1 f 2 - f 1 = M a . . . 2 In Wheel 2 , a torque τ is applied due to wheel and gear, so, τ - f 2 R = I α = I a R . . . 3 In Wheel 1 , f 1 R = I α = I a R . . . 4 From equation 2 , 3 and 4 , a = τ R M R 2 + 2 I (c) When the acceleration will become maximum, the torque on Wheel 2 will be maximum and the friction on it will also be maximum, i.e., f 2 = μ N 2 , put this in all above equations, μ N 2 - f 1 = M a m . . . 5 τ m - μ N 2 R = I a m R . . . 6 f 1 R = I a m R . . . 7 a m = τ m R M R 2 + 2 I . . . 8 The net torque about the centre of mass will be zero, so, ∑ τ C M = 0 N 2 2 R + f 1 R = N 1 2 R + μ N 2 R 2 N 2 + f 1 = 2 N 1 + μ N 2 . . . 9 From equation 1 , 2 N 2 + f 1 = 2 M g - N 2 + μ N 2 4 N 2 + f 1 = 2 M g + μ N 2 From equation 5 , 4 N 2 = 2 M g + M a m . . . 10 From equation 6 and 10 , τ m = a m M R + 2 I R . . . 11 From equation 8 and 11 , a m = μ g 2 1 - μ 4 (d) For μ = 1 , a m = 1 g 2 1 - 1 4 = 2 g 3 .

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Calculate the moment of inertia of a uniform solid cone relative to its symmetry axis, if the mass of the cone is equal to m and the radius of its base to $R$ .

Important Questions on Rotational Mechanics

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 4>Q 23

A force

\vec{F} = A \hat{i} + B \hat{j}

is applied to a point whose radius vector, relative to the origin of coordinates

O

, is equal to

\vec{r} = a \hat{i} + b \hat{j}

, where

a

b

A

and

B

are constants, and

\hat{i}

and

d m

are the unit vectors along

x

and

y

-axes. Find the moment

N

and the arm

l

of the force

F

, relative to the point

O

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 4>Q 24

A uniform cylinder of radius $R$ and mass $M$ can rotate freely about a stationary horizontal axis $O$ . A thin cord of length $l$ and mass $m$ is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as a function of the length $x$ of the hanging part of the cord. The wound part of the cord is supposed to have its centre of gravity on the cylinder axis.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 4>Q 25

A vertically oriented uniform rod, of mass

M

and length

l

, can rotate about its upper end. A horizontally flying bullet, of mass

m

, strikes the lower end of the rod and gets stuck on it; as a result, the rod swings through an angle

α

. Assuming that

m ≪ M,

find

(a)

the velocity of the flying bullet

(b)

the momentum increment in the system "bullet-rod" during the impact; what causes the change of that momentum

(c)

at what distance

x

, from the upper end of the rod, the bullet must strike, for the momentum of the system "bullet-rod" to remain constant during the impact.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 4>Q 26

A point $A$ is located on the rim of a wheel of radius $R = 0.50 m$ which rolls without slipping along a horizontal surface with velocity $v = 1.00 m / s$ as shown in figure. Find:

(a) the modulus and the direction of the acceleration vector of the point $A$ ;

(b) the total distance s traversed by the point $A$ between the two successive moments at which it touches the surface.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 4>Q 27

A uniform sphere of mass $m$ and radius $r$ rolls without sliding over a horizontal plane, rotating about a horizontal axle $OA$ . In the process, the centre of the sphere moves with velocity along a circle of radius $R$ Find the kinetic energy of the sphere.

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Alpha Question Bank for Engineering: Physics>Chapter 10 - Rotational Mechanics>Exercise - 4>Q 30

Consider a bicycle in vertical position accelerating forward without slipping on a straight horizontal road. The combined mass of the bicycle and the rider is $M$ and the magnitude of the accelerating torque applied on the rear wheel by the pedal and gear system is $τ$ . The radius and the moment of inertia of each wheel is $R$ and $I$ (with respect to the axis) respectively. The acceleration due to gravity is $g$ .

(a) Draw the free diagram of the system (bicycle and rider ).

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(b) Obtains the acceleration $a$ in terms of the above-mentioned quantities. $a =$

(c) For simplicity assume that the centre of mass of the system is at height $R$ from the ground and equidistant at $2 R$ from the centre of each of the wheels. Let $μ$ be the coefficient of friction (both static and dynamic) between the wheels and the ground. Consider $M > > \frac{I}{R^{2}}$ and no slipping. Obtain the conditions for the maximum acceleration $a_{m}$ of the bike. $a_{m} =$

(d) For $μ = 1.0$ calculate $a_{m}$ . $a_{m} =$

Calculate the moment of inertia of a uniform solid cone relative to its symmetry axis, if the mass of the cone is equal to m and the radius of its base to R.

Important Questions on Rotational Mechanics

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Calculate the moment of inertia of a uniform solid cone relative to its symmetry axis, if the mass of the cone is equal to m and the radius of its base to $R$ .