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10 months ago

Physics System of Particles and Rotational Motion

A disc of radius R is rotating with an angular speed ωo about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is µk.

(a) What was the velocity of its centre of mass before being brought in contact with the table?

(b) What happens to the linear velocity of a point on its rim when placed in contact with the table? (c) What happens to the linear speed of the centre of mass when disc is placed in contact with the table?

(d) Which force is responsible for the effects in (b) and (c).

(e) What condition should be satisfied for rolling to begin?

(f) Calculate the time taken for the rolling to begin

...more

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Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

(a) Before being bought in contact with the table the disc was in pure rotational motion. Hence V_cm=0

(b) when the disc is placed in contact with table due to friction centre of mass acquires some linear velocity.

(c) when the rotating disc is placed in contact with the table due to friction centre of mass acquires some linear velocity.

(d) friction is responsible for the effects ib b and c

(e) when rolling starts V_cm=wR

(f) time period for rolling to begin is t= $\frac{R w_{o}}{μ g (1 + \frac{m R^{2}}{I})}$

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New answer posted

10 months ago

Physics System of Particles and Rotational Motion

Two discs of moments of inertia I₁ and I₂ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed ω₁ and w₂ are brought into contact face to face with their axes of rotation coincident.

(a) Does the law of conservation of angular momentum apply to the situation? why?

(b) Find the angular speed of the two-disc system.

(c) Calculate the loss in kinetic energy of the system in the process.

(d) Account for this loss.

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Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

(a) yes the law of conservation of angular momentum can be applied because there is no net external torque on the system of the two discs.

External forces gravitation and normal reaction act through the axis of rotation, hence produce no torque.

(b) by conservation of angular momentum

L_f= L_i

Iw=I₁w₁+I₂w₂

So w= $\frac{I_{1} w_{1} + I_{2} w_{2}}{I_{1} + I_{2}}$

(c) K_f= $\frac{1}{2} (I_{1} + I_{2}) \frac{{(I_{1} w_{1} + I_{2} w_{2)}}^{2}}{I_{1} + I_{2}}$

K_f= ½ (I₁w₁²+I₂w₂²)

$? k =$ K_f-K_i =- $\frac{- I_{1} I_{2}}{2 (I_{1} + I_{2})} (w_{1} - w_{2})$ ²<0

(d) hence there is loss of KE of the system. The loss of kinetic energy is mainly due to work against the friction.

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New answer posted

10 months ago

Physics System of Particles and Rotational Motion

Find the centre of mass of a uniform (a) half-disc, (b) quarter-disc.

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Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

let M and R be mass and radius of half disc , mass per unit area of half disc

So m = $\frac{M}{\frac{1}{2} π R^{2}}$

(a) the half disc be supposed to be consist of a large number of semicircular ring of mass dm and thickness dr and radii ranging from r=0 to r=R.

surface area of semicircular ring of radius r and thickness dr = $\frac{1}{2} 2 π r d r = π r d r$

so mass of the elementary ring dm = $π r d r * \frac{2 M}{π R^{2}}$

dm= $\frac{2 M}{R^{2}} r d r$

if x,y are coordinates of centre of mass of this element,

then (x,y)=(0,2r/ $π$ )

so x=0 and y =2r/ $π$

let x_cm and y_cm be the coordinates of the centre of mass of the semicircular disc

so x_cm= $\frac{1}{M} \int_{o}^{R} x d m = \frac{1}{M} \int_{o}^{R} 0 d m = 0$

Y_cm= $\frac{1}{M} \int_{o}^{R} y d m = \frac{1}{M} \int_{0}^{R} \frac{2 r}{π} * (\frac{2 M}{R^{2}} r d r)$

= $\frac{4}{π R^{2}} \int_{0}^{R} r^{2} d r = \frac{4}{π R^{2}} [\frac{r^{3}}{3}]$ ₀^R

= 4R/3 $π$

Centre of mass of a uniform quar

...more

This is a long answer type question as classified in NCERT Exemplar

let M and R be mass and radius of half disc , mass per unit area of half disc

So m = $\frac{M}{\frac{1}{2} π R^{2}}$

(a) the half disc be supposed to be consist of a large number of semicircular ring of mass dm and thickness dr and radii ranging from r=0 to r=R.

surface area of semicircular ring of radius r and thickness dr = $\frac{1}{2} 2 π r d r = π r d r$

so mass of the elementary ring dm = $π r d r * \frac{2 M}{π R^{2}}$

dm= $\frac{2 M}{R^{2}} r d r$

if x,y are coordinates of centre of mass of this element,

then (x,y)=(0,2r/ $π$ )

so x=0 and y =2r/ $π$

let x_cm and y_cm be the coordinates of the centre of mass of the semicircular disc

so x_cm= $\frac{1}{M} \int_{o}^{R} x d m = \frac{1}{M} \int_{o}^{R} 0 d m = 0$

Y_cm= $\frac{1}{M} \int_{o}^{R} y d m = \frac{1}{M} \int_{0}^{R} \frac{2 r}{π} * (\frac{2 M}{R^{2}} r d r)$

= $\frac{4}{π R^{2}} \int_{0}^{R} r^{2} d r = \frac{4}{π R^{2}} [\frac{r^{3}}{3}]$ ₀^R

= 4R/3 $π$

Centre of mass of a uniform quarter disc

Mass per unit area of the quarter disc = $\frac{M}{\frac{π R^{2}}{4}} = \frac{4 M}{π R^{2}}$

For half disc along yaxis and xaxis =4R/3 $π$

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New question posted

10 months ago

Physics Semiconductor Electronics

Q.6. Let $A = [- 1, 1]$ , Then, discuss whether the following functions defined on $A$ are one-one, onto, or bijective:

(i) $f (x) = \frac{x}{2}$

(ii) $g (x) = | x |$

(iii) $h (x) = x | x |$

(iv) $k (x) = x^{2}$ .

0 Follower 9 Views

New question posted

10 months ago

Physics Semiconductor Electronics

Q.6. Let $A = [- 1, 1]$ , Then, discuss whether the following functions defined on $A$ are one-one, onto, or bijective:

(i) $f (x) = \frac{x}{2}$

(ii) $g (x) = | x |$

(iii) $h (x) = x | x |$

(iv) $k (x) = x^{2}$ .

0 Follower 7 Views

New question posted

10 months ago

Physics Systems of Particles and Rotational Motion

Q.6. Let $A = [- 1, 1]$ , Then, discuss whether the following functions defined on $A$ are one-one, onto, or bijective:

(i) $f (x) = \frac{x}{2}$

(ii) $g (x) = | x |$

(iii) $h (x) = x | x |$

(iv) $k (x) = x^{2}$ .

0 Follower 4 Views

New question posted

10 months ago

Physics Systems of Particles and Rotational Motion

Q.6. Let $A = [- 1, 1]$ , Then, discuss whether the following functions defined on $A$ are one-one, onto, or bijective:

(i) $f (x) = \frac{x}{2}$

(ii) $g (x) = | x |$

(iii) $h (x) = x | x |$

(iv) $k (x) = x^{2}$ .

0 Follower 3 Views

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