A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure. A symmetric system consisting of a disc (top), solid sphere (bottom-left), and spherical shell (bottom-right) arranged with vertical axis PQ. The moment of inertia of the given system about PQ is fracmathrmx15mathrmI,$\frac{\mathrm{x}}{15}\mathrm{I},$ where I is the moment of inertia of the disc about its diameter. The value of x is
Numerical Answer Type:
Enter a numerical valueAnswer: 199 to 199+4 marks
Solution & Explanation
### Related Formula
Parallel Axis Theorem:
I_textaxis = I_textcom + M R^2$$I_{\text{axis}} = I_{\text{com}} + M R^2$$
Standard Moments of Inertia about center of mass:
- Disc about diameter: I_textdisc,dia = fracMR^24$I_{\text{disc,dia}} = \frac{MR^2}{4}$
- Solid sphere: I_textsphere = frac25MR^2$I_{\text{sphere}} = \frac{2}{5}MR^2$
- Spherical shell: I_textshell = frac23MR^2$I_{\text{shell}} = \frac{2}{3}MR^2$
### Core Logic
The axis of rotation PQ passes through the center of the top disc (A) along its diameter.
- Top disc (A):
I_A = fracMR^24$$I_A = \frac{MR^2}{4}$$
- Bottom-left solid sphere (B): Center lies at distance R$R$ from the axis PQ.
I_B = I_textcom + M R^2 = frac25MR^2 + MR^2 = frac75MR^2$$I_B = I_{\text{com}} + M R^2 = \frac{2}{5}MR^2 + MR^2 = \frac{7}{5}MR^2$$
- Bottom-right spherical shell (C): Center lies at distance R$R$ from the axis PQ.
I_C = I_textcom + M R^2 = frac23MR^2 + MR^2 = frac53MR^2$$I_C = I_{\text{com}} + M R^2 = \frac{2}{3}MR^2 + MR^2 = \frac{5}{3}MR^2$$
### Step 1: Calculate Total System Moment of Inertia
Sum the contributions:
I_textPQ = I_A + I_B + I_C$$I_{\text{PQ}} = I_A + I_B + I_C$$I_textPQ = fracMR^24 + frac75MR^2 + frac53MR^2$$I_{\text{PQ}} = \frac{MR^2}{4} + \frac{7}{5}MR^2 + \frac{5}{3}MR^2$$
To add the fractions, find a common denominator (60$60$):
I_textPQ = left( frac15 + 84 + 10060 right) MR^2 = frac19960 MR^2$$I_{\text{PQ}} = \left( \frac{15 + 84 + 100}{60} \right) MR^2 = \frac{199}{60} MR^2$$
### Step 2: Express in terms of standard Disc Moment
We are given I = fracMR^24 implies MR^2 = 4I$I = \frac{MR^2}{4} \implies MR^2 = 4I$. Substitute this in the expression:
I_textPQ = frac19960 (4I) = frac19915 I$$I_{\text{PQ}} = \frac{199}{60} (4I) = \frac{199}{15} I$$
Comparing with I_textPQ = fracx15 I$I_{\text{PQ}} = \frac{x}{15} I$ yields x = 199$x = 199$.
### Pattern Recognition
Sees: Composite body consisting of three standard symmetric shapes about a tangent/offset axis.
Shortcut: Sum the central inertia terms and the offset terms separately. Offset masses are only B and C, so the offset sum is 2MR^2$2MR^2$. The central sum is (1/4 + 2/5 + 2/3)MR^2$(1/4 + 2/5 + 2/3)MR^2$. Adding these directly yields the combined fractional factor of 199/60$199/60$.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: System of Particles and Rotational Motion
A symmetric system consisting of a disc (top), solid sphere (bottom-left), and spherical shell (bottom-right) arranged with vertical axis PQ.
Keywords:#moment of inertia of disc sphere shell system#JEE Main 2025 Morning Q21#Rotational Dynamics JEE Main 2025#parallel axis theorem composite shapes#moment of inertia#parallel axis theorem#disc#solid sphere#spherical shell
More System of Particles and Rotational Motion Previous-Year Questions — Page 6
Q8jee_main_2025_24_jan_morningAngular Momentum
An object of mass 'm' is projected from origin in a vertical xy plane at an angle 45^circ$45^{\circ}$ with the x-axis with an initial velocity v_0$v_{0}$ The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be
[g is acceleration due to gravity]
A.fracmv_0^32sqrt2g$\frac{mv_{0}^{3}}{2\sqrt{2}g}$ along negative z-axis
B.fracmv_0^32sqrt2g$\frac{mv_{0}^{3}}{2\sqrt{2}g}$ along positive z-axis
C.fracmv_0^34sqrt2g$\frac{mv_{0}^{3}}{4\sqrt{2}g}$ along positive z-axis
D.fracmv_0^34sqrt2g$\frac{mv_{0}^{3}}{4\sqrt{2}g}$ along negative z-axis
Solution
### Related Formula
The definition of angular momentum vector vecL$\vec{L}$ relative to the origin is:
vecL = vecr times vecp = m(vecr times vecv) $$\vec{L} = \vec{r} \times \vec{p} = m(\vec{r} \times \vec{v}) $$
In scalar form for a horizontal speed component at a maximum altitude H$H$:
L = m v_x H$L = m v_{x} H$
### Core Logic
At maximum height, the vertical speed component drops to zero, and the projectile travels entirely horizontally along the curve profile as shown in Angular Momentum diagram for Q8 - JEE Main 2025 Morning :
v_x = v_0 cos 45^circ = fracv_0sqrt2$$v_{x} = v_{0} \cos 45^{\circ} = \frac{v_{0}}{\sqrt{2}}$$H = fracv_0^2 sin^2 45^circ2g = fracv_0^24g$$H = \frac{v_{0}^{2} \sin^{2} 45^{\circ}}{2g} = \frac{v_{0}^{2}}{4g}$$
### Step 1: Calculating Magnitude and Vector Direction
Evaluate the horizontal vector cross components:
L = m left(fracv_0sqrt2
ight) left(fracv_0^24g
ight) = fracmv_0^34sqrt2g$$L = m \left(\frac{v_{0}}{\sqrt{2}}
ight) \left(\frac{v_{0}^{2}}{4g}
ight) = \frac{mv_{0}^{3}}{4\sqrt{2}g}$$
Using the right-hand rule, vecr$\vec{r}$ points into quadrant-1 while velocity points towards +hati$+\hat{i}$. Therefore, vecr times vecv$\vec{r} \times \vec{v}$ tracks clockwise, yielding a negative hatk$\hat{k}$ orientation (along the negative z-axis).
### Pattern Recognition
At peak height, always map vecL$\vec{L}$ using m cdot v_textpeak cdot y_textmax$m \cdot v_{\text{peak}} \cdot y_{\text{\max}}$. This scalar form simplifies the calculation significantly.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: System of Particles and Rotational Motion
Q11jee_main_2025_24_jan_morningRolling Motion
A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45^circ$45^{\circ}$ If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be :-
A.frac1sqrt2 g$\frac{1}{\sqrt{2}} g$
B.frac13sqrt2 g$\frac{1}{3\sqrt{2}} g$
C.fracsqrt2 g3$\frac{\sqrt{2} g}{3}$
D.sqrt2 g$\sqrt{2} g$
Solution
### Related Formula
The linear acceleration a$a$ for pure rolling motion down an incline profile is:
a = fracgsintheta1 + fracImr^2$$a = \frac{g\sin\theta}{1 + \frac{I}{mr^{2}}}$$
### Core Logic
For a uniform solid cylinder, the moment of inertia around its central axis is:
I = frac12mr^2 implies fracImr^2 = frac12$$I = \frac{1}{2}mr^{2} \implies \frac{I}{mr^{2}} = \frac{1}{2}$$
### Step 1: Calculating Acceleration
Substitute theta = 45^circ$\theta = 45^{\circ}$ and the cylinder inertial factor into the formula :
a = fracgsin 45^circ1 + frac12 = fracfracgsqrt2frac32$$a = \frac{g\sin 45^{\circ}}{1 + \frac{1}{2}} = \frac{\frac{g}{\sqrt{2}}}{\frac{3}{2}}$$a = frac2g3sqrt2 = fracsqrt2g3$$a = \frac{2g}{3\sqrt{2}} = \frac{\sqrt{2}g}{3}$$
### Pattern Recognition
Solid cylinder rolls with acceleration matching frac23 gsintheta$\frac{2}{3} g\sin\theta$. Since sin 45^circ = frac1sqrt2$\sin 45^{\circ} = \frac{1}{\sqrt{2}}$, this simplifies directly to fracsqrt2g3$\frac{\sqrt{2}g}{3}$.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: System of Particles and Rotational Motion
Q17jee_main_2025_28_jan_eveningTorque and Equilibrium
A uniform rod of mass 250mathrmg$250\mathrm{g}$ having length 100mathrmcm$100\mathrm{cm}$ is balanced on a sharp edge at 40mathrmcm$40\mathrm{cm}$ mark[cite: 150, 151]. A mass of 400mathrmg$400\mathrm{g}$ is suspended at 10mathrmcm$10\mathrm{cm}$ mark. To maintain the balance of the rod, the mass to be suspended at 90mathrmcm$90\mathrm{cm}$ mark, is [cite: 154, 156]
A.300mathrmg$300\mathrm{g}$
B.190mathrmg$190\mathrm{g}$
C.200mathrmg$200\mathrm{g}$
D.290mathrmg$290\mathrm{g}$
Solution
### Related Formula
For rotational equilibrium, the \sum of all counter-clockwise torques about the pivot point must exactly balance the \sum of all clockwise torques:
sum tau_textpivot = 0 implies sum (m_i cdot g cdot x_i) = 0$$\sum \tau_{\text{pivot}} = 0 \implies \sum (m_i \cdot g \cdot x_i) = 0$$
### Core Logic
The rod is uniform, meaning its mass (250text g$250\text{ g}$) acts exactly at its geometric center of mass, the 50text cm$50\text{ cm}$ mark[cite: 150, 151]. Let the pivot point be the sharp edge at the 40text cm$40\text{ cm}$ mark .
Calculate the relative lever arms from the pivot [cite: 775, 776, 777]:
* 400text g$400\text{ g}$ mass at 10text cm$10\text{ cm}$ mark: lever arm = 40 - 10 = 30text cm$= 40 - 10 = 30\text{ cm}$ (counter-clockwise)
* 250text g$250\text{ g}$ rod mass at 50text cm$50\text{ cm}$ mark: lever arm = 50 - 40 = 10text cm$= 50 - 40 = 10\text{ cm}$ (clockwise)
* Unknown mass M$M$ at 90text cm$90\text{ cm}$ mark: lever arm = 90 - 40 = 50text cm$= 90 - 40 = 50\text{ cm}$ (clockwise)
Setting up the torque balance equation:
400 times 30 = (250 times 10) + (M times 50)$$400 \times 30 = (250 \times 10) + (M \times 50)$$12000 = 2500 + 50M$$12000 = 2500 + 50M$$50M = 9500 implies M = frac950050 = 190text g$$50M = 9500 \implies M = \frac{9500}{50} = 190\text{ g}$$
### Step 1: Visual Context
The structural layout of forces acting on the balanced rod system is shown below:
Torque and Equilibrium balancing diagram for Q17
### Pattern Recognition
Never forget to include the weight of a uniform rod itself in equilibrium equations. It is a common oversight to omit the rod's mass, which always acts at its geometric center.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: System of Particles and Rotational Motion
Qjee_main_2025_29_jan_morningTorque
The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of vecmathrmF = hatmathrmi -hatmathrmj +hatmathrmk$\vec{\mathrm{F}} = \hat{\mathrm{i}} -\hat{\mathrm{j}} +\hat{\mathrm{k}}$ acts on the particle, then the magnitude of torque (with respect to origin) in z$z$ -direction is
Numerical Answer.Answer: 2 to 2
Solution
### Related Formula
vectau = vecr times vecF$$\vec{\tau} = \vec{r} \times \vec{F}$$
### Core Logic
Given position vector vecr = hati + hatj + hatk$\vec{r} = \hat{i} + \hat{j} + \hat{k}$ and force vecF = hati - hatj + hatk$\vec{F} = \hat{i} - \hat{j} + \hat{k}$:
vectau = left| beginarrayccc hati & hatj & hatk \\ 1 & 1 & 1 \\ 1 & -1 & 1 endarray right|$$\vec{\tau} = \left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{array} \right|$$
### Step 1: Isolate z-component
tau_z = hatk(1(-1) - 1(1)) = -2hatk$$\tau_z = \hat{k}(1(-1) - 1(1)) = -2\hat{k}$$
The absolute magnitude of the torque component in the z$z$-direction equals 2 mathrm~N cdot m$2 \mathrm{~N \cdot m}$.
### Chapter Mix
Class 11 Physics: System of Particles and Rotational Motion
Q56jee_main_2024_01_february_morningCentre of Mass
The identical spheres each of mass 2M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 4mathrm~m$4\mathrm{~m}$ each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is frac4sqrt2x$\frac{4\sqrt{2}}{x}$, where the value of x$x$ is ______.
Numerical Answer.Answer: 3 to 3
Solution
### Related Formula
Position vector of the Centre of Mass (COM):
vecr_textCOM = fracm_1vecr_1 + m_2vecr_2 + m_3vecr_3m_1 + m_2 + m_3$$\vec{r}_{\text{COM}} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + m_3\vec{r}_3}{m_1 + m_2 + m_3}$$
### Core Logic
Assign coordinates to the three masses (m_1=m_2=m_3=2M$m_1=m_2=m_3=2M$):
- Origin mass: vecr_1 = 0hati + 0hatj$\vec{r}_1 = 0\hat{i} + 0\hat{j}$
- X-axis mass: vecr_2 = 4hati + 0hatj$\vec{r}_2 = 4\hat{i} + 0\hat{j}$
- Y-axis mass: vecr_3 = 0hati + 4hatj$\vec{r}_3 = 0\hat{i} + 4\hat{j}$
Substitute these into the COM formula:
vecr_textCOM = frac2M(0) + 2M(4hati) + 2M(4hatj)2M + 2M + 2M = frac8Mhati + 8Mhatj6M = frac43hati + frac43hatj$$\vec{r}_{\text{COM}} = \frac{2M(0) + 2M(4\hat{i}) + 2M(4\hat{j})}{2M + 2M + 2M} = \frac{8M\hat{i} + 8M\hat{j}}{6M} = \frac{4}{3}\hat{i} + \frac{4}{3}\hat{j}$$
### Step 1: Calculate Position Vector Magnitude
|vecr_textCOM| = sqrtleft(frac43right)^2 + left(frac43right)^2 = frac4sqrt23$$|\vec{r}_{\text{COM}}| = \sqrt{\left(\frac{4}{3}\right)^2 + \left(\frac{4}{3}\right)^2} = \frac{4\sqrt{2}}{3}$$
Matching this directly with the given template frac4sqrt2x$\frac{4\sqrt{2}}{x}$ shows that x = 3$x = 3$.
### Pattern Recognition
Since the mass layout is completely symmetric along both right-angle legs, the COM coordinates are identical (x_textCOM = y_textCOM$x_{\text{COM}} = y_{\text{COM}}$).
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Physics: System of Particles and Rotational Motion
More System of Particles and Rotational Motion Questions — jee_main_2025_07_april_morning
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