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Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a?
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Uniform solid disk of mass M and radius R is rolling without slipping ...
Free Body Diagrams
To analyze the system, we will consider the forces acting on the block and the disk separately.
- For the block of mass m:
- Weight (mg) acting downwards.
- Tension (T) acting upwards.
- For the disk (mass M, radius R):
- Tension (T) acting tangentially at the radius (R).
- The frictional force (f) acting horizontally to prevent slipping.
Equations of Motion
1. **For the block (Newton’s Second Law)**:
\[
mg - T = ma \quad \text{(1)}
\]
2. **For the disk (Newton’s Second Law)**:
The net torque (\( \tau \)) on the disk caused by tension is given by:
\[
\tau = T \cdot R
\]
The angular acceleration (\( \alpha \)) is related to linear acceleration (\( a \)) by:
\[
\alpha = \frac{a}{R} \quad \text{(2)}
\]
Using the rotational form of Newton's second law:
\[
T \cdot R = I \cdot \alpha \quad \text{(3)}
\]
For a solid disk, the moment of inertia \( I \) is:
\[
I = \frac{1}{2} M R^2
\]
Substituting \( \alpha \) from equation (2):
\[
T \cdot R = \frac{1}{2} M R^2 \cdot \frac{a}{R}
\]
Simplifying gives:
\[
T = \frac{1}{2} M a \quad \text{(4)}
\]
Solving the System
Substituting equation (4) into equation (1):
\[
mg - \frac{1}{2} M a = ma
\]
Combining terms:
\[
mg = ma + \frac{1}{2} M a
\]
\[
mg = a\left(m + \frac{1}{2} M\right)
\]
Thus, the linear acceleration \( a \) of the block is:
\[
a = \frac{mg}{m + \frac{1}{2} M}
\]
Finding Angular Acceleration
Using equation (2):
\[
\alpha = \frac{a}{R} = \frac{mg}{R(m + \frac{1}{2}M)}
\]
Finding Tension
Substituting \( a \) back into equation (4):
\[
T = \frac{1}{2} M \left(\frac{mg}{m + \frac{1}{2} M}\right)
\]
This detailed analysis yields the linear acceleration \( a \), angular acceleration \( \alpha \), and tension \( T \) in the system.
To analyze the system, we will consider the forces acting on the block and the disk separately.
- For the block of mass m:
- Weight (mg) acting downwards.
- Tension (T) acting upwards.
- For the disk (mass M, radius R):
- Tension (T) acting tangentially at the radius (R).
- The frictional force (f) acting horizontally to prevent slipping.
Equations of Motion
1. **For the block (Newton’s Second Law)**:
\[
mg - T = ma \quad \text{(1)}
\]
2. **For the disk (Newton’s Second Law)**:
The net torque (\( \tau \)) on the disk caused by tension is given by:
\[
\tau = T \cdot R
\]
The angular acceleration (\( \alpha \)) is related to linear acceleration (\( a \)) by:
\[
\alpha = \frac{a}{R} \quad \text{(2)}
\]
Using the rotational form of Newton's second law:
\[
T \cdot R = I \cdot \alpha \quad \text{(3)}
\]
For a solid disk, the moment of inertia \( I \) is:
\[
I = \frac{1}{2} M R^2
\]
Substituting \( \alpha \) from equation (2):
\[
T \cdot R = \frac{1}{2} M R^2 \cdot \frac{a}{R}
\]
Simplifying gives:
\[
T = \frac{1}{2} M a \quad \text{(4)}
\]
Solving the System
Substituting equation (4) into equation (1):
\[
mg - \frac{1}{2} M a = ma
\]
Combining terms:
\[
mg = ma + \frac{1}{2} M a
\]
\[
mg = a\left(m + \frac{1}{2} M\right)
\]
Thus, the linear acceleration \( a \) of the block is:
\[
a = \frac{mg}{m + \frac{1}{2} M}
\]
Finding Angular Acceleration
Using equation (2):
\[
\alpha = \frac{a}{R} = \frac{mg}{R(m + \frac{1}{2}M)}
\]
Finding Tension
Substituting \( a \) back into equation (4):
\[
T = \frac{1}{2} M \left(\frac{mg}{m + \frac{1}{2} M}\right)
\]
This detailed analysis yields the linear acceleration \( a \), angular acceleration \( \alpha \), and tension \( T \) in the system.
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Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a?
Question Description
Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a?.
Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a?.
Solutions for Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? in English & in Hindi are available as part of our courses for JEE.
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Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a?, a detailed solution for Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? has been provided alongside types of Uniform solid disk of mass M and radius R is rolling without slipping on a horizontal surface. The disk is connected to a light inextensible string wound around its circumference. The string passes over a small, frictionless pulley fixed at the edge of a horizontal table and is connected to a block of mass m hanging vertically over the edge of the table. The system is released from rest. Assuming no slipping occurs between the disk and the surface, find the linear acceleration a of the block, the angular acceleration o of the disk, and the tension T in the string. Options: A) The linear acceleration of the block is a B) The angular acceleration of the disk is a C) The tension in the string is T D) The linear acceleration of the block is a? theory, EduRev gives you an
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