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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.?
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A uniform solid disk of mass M and radius R is rolling without slippin...
Free Body Diagram Analysis
To analyze the system, consider the forces acting on the block and the disk. The block of mass \( m \) experiences gravitational force \( mg \) downward and tension \( T \) upward. The disk of mass \( M \) rolls without slipping.
Equations of Motion
- For the block:
\[
mg - T = ma \quad (1)
\]
- For the disk:
The net torque \( \tau \) on the disk due to the tension \( T \) is given by:
\[
\tau = TR = I \alpha \quad (2)
\]
where \( I = \frac{1}{2} MR^2 \) (moment of inertia of the disk) and \( \alpha \) is the angular acceleration.
Since the disk rolls without slipping, we have the relationship:
\[
a = R \alpha \quad (3)
\]
Substituting and Solving
From equation (3), we can express \( \alpha \):
\[
\alpha = \frac{a}{R}
\]
Substituting \( \alpha \) into equation (2):
\[
TR = \frac{1}{2} MR^2 \cdot \frac{a}{R}
\]
\[
T = \frac{1}{2} Ma \quad (4)
\]
Now substitute \( T \) from equation (4) into equation (1):
\[
mg - \frac{1}{2} Ma = ma
\]
\[
mg = ma + \frac{1}{2} Ma
\]
\[
mg = a \left(m + \frac{1}{2} M\right)
\]
\[
a = \frac{mg}{m + \frac{1}{2}M} \quad (5)
\]
Finding Angular Acceleration and Tension
Using equation (3) to find \( \alpha \):
\[
\alpha = \frac{a}{R} = \frac{mg}{R(m + \frac{1}{2}M)} \quad (6)
\]
Finally, substitute \( a \) from equation (5) into equation (4) to find \( T \):
\[
T = \frac{1}{2} M \left( \frac{mg}{m + \frac{1}{2}M} \right) \quad (7)
\]
Conclusion
- The linear acceleration \( a \) of the block is given by equation (5).
- The angular acceleration \( \alpha \) of the disk is given by equation (6).
- The tension \( T \) in the string is found using equation (7).
To analyze the system, consider the forces acting on the block and the disk. The block of mass \( m \) experiences gravitational force \( mg \) downward and tension \( T \) upward. The disk of mass \( M \) rolls without slipping.
Equations of Motion
- For the block:
\[
mg - T = ma \quad (1)
\]
- For the disk:
The net torque \( \tau \) on the disk due to the tension \( T \) is given by:
\[
\tau = TR = I \alpha \quad (2)
\]
where \( I = \frac{1}{2} MR^2 \) (moment of inertia of the disk) and \( \alpha \) is the angular acceleration.
Since the disk rolls without slipping, we have the relationship:
\[
a = R \alpha \quad (3)
\]
Substituting and Solving
From equation (3), we can express \( \alpha \):
\[
\alpha = \frac{a}{R}
\]
Substituting \( \alpha \) into equation (2):
\[
TR = \frac{1}{2} MR^2 \cdot \frac{a}{R}
\]
\[
T = \frac{1}{2} Ma \quad (4)
\]
Now substitute \( T \) from equation (4) into equation (1):
\[
mg - \frac{1}{2} Ma = ma
\]
\[
mg = ma + \frac{1}{2} Ma
\]
\[
mg = a \left(m + \frac{1}{2} M\right)
\]
\[
a = \frac{mg}{m + \frac{1}{2}M} \quad (5)
\]
Finding Angular Acceleration and Tension
Using equation (3) to find \( \alpha \):
\[
\alpha = \frac{a}{R} = \frac{mg}{R(m + \frac{1}{2}M)} \quad (6)
\]
Finally, substitute \( a \) from equation (5) into equation (4) to find \( T \):
\[
T = \frac{1}{2} M \left( \frac{mg}{m + \frac{1}{2}M} \right) \quad (7)
\]
Conclusion
- The linear acceleration \( a \) of the block is given by equation (5).
- The angular acceleration \( \alpha \) of the disk is given by equation (6).
- The tension \( T \) in the string is found using equation (7).
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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.?
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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.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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.?.
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.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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.?.
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