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Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?
- a)Both cylinders P and Q reach the ground at the same time.
- b)Cylinders P has larger linear acceleration than cylinder Q.
- c)Both cylinders reach the ground with same translational kinetic energy.
- d)Cylinder Q reaches the ground with larger angular speed.
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two solid cylinders P and Q of same mass and same radius start rolling...
The acceleration of the center of mass of cylinder rolling down an inclined plane is
Here IP > IQ because in case of P the mass is concentrated away from the axis. 
Most Upvoted Answer
Two solid cylinders P and Q of same mass and same radius start rolling...
Explanation:
To understand why option D is correct, let us analyze the motion of the two cylinders P and Q.
Rolling Motion:
When a cylinder rolls without slipping, it undergoes both translational and rotational motion. The motion can be described by the following equations:
1. Translational Motion:
The linear acceleration of a rolling cylinder can be determined using the equation:
a = g * sin(θ) * (1 - k^2)
where a is the linear acceleration, g is the acceleration due to gravity, θ is the angle of inclination of the plane, and k is the radius of gyration of the cylinder.
2. Rotational Motion:
The angular acceleration of a rolling cylinder can be determined using the equation:
α = g * sin(θ) * k / R
where α is the angular acceleration and R is the radius of the cylinder.
Comparison of Cylinders P and Q:
Cylinder P has most of its mass concentrated near its surface, which means it has a smaller radius of gyration (k) compared to cylinder Q. On the other hand, cylinder Q has most of its mass concentrated near the axis, which means it has a larger radius of gyration.
Now, let's analyze the statements given in the question:
a) Both cylinders P and Q reach the ground at the same time:
This statement is incorrect. Since cylinder P has a larger linear acceleration than cylinder Q (due to its smaller radius of gyration), it will reach the ground before cylinder Q.
b) Cylinder P has larger linear acceleration than cylinder Q:
This statement is correct. As mentioned above, cylinder P has a larger linear acceleration due to its smaller radius of gyration.
c) Both cylinders reach the ground with the same translational kinetic energy:
This statement is incorrect. The translational kinetic energy of a rolling cylinder is given by the equation:
KE_trans = (1/2) * M * v^2
where M is the mass of the cylinder and v is its linear velocity. Since cylinder P reaches the ground with a larger linear velocity (due to its larger linear acceleration), it will have a greater translational kinetic energy compared to cylinder Q.
d) Cylinder Q reaches the ground with a larger angular speed:
This statement is correct. The angular speed of a rolling cylinder is given by the equation:
ω = v / R
where ω is the angular speed and R is the radius of the cylinder. Since cylinder P reaches the ground with a larger linear velocity (due to its larger linear acceleration), it will have a greater angular speed compared to cylinder Q.
Therefore, the correct statement is option D - Cylinder Q reaches the ground with a larger angular speed.
To understand why option D is correct, let us analyze the motion of the two cylinders P and Q.
Rolling Motion:
When a cylinder rolls without slipping, it undergoes both translational and rotational motion. The motion can be described by the following equations:
1. Translational Motion:
The linear acceleration of a rolling cylinder can be determined using the equation:
a = g * sin(θ) * (1 - k^2)
where a is the linear acceleration, g is the acceleration due to gravity, θ is the angle of inclination of the plane, and k is the radius of gyration of the cylinder.
2. Rotational Motion:
The angular acceleration of a rolling cylinder can be determined using the equation:
α = g * sin(θ) * k / R
where α is the angular acceleration and R is the radius of the cylinder.
Comparison of Cylinders P and Q:
Cylinder P has most of its mass concentrated near its surface, which means it has a smaller radius of gyration (k) compared to cylinder Q. On the other hand, cylinder Q has most of its mass concentrated near the axis, which means it has a larger radius of gyration.
Now, let's analyze the statements given in the question:
a) Both cylinders P and Q reach the ground at the same time:
This statement is incorrect. Since cylinder P has a larger linear acceleration than cylinder Q (due to its smaller radius of gyration), it will reach the ground before cylinder Q.
b) Cylinder P has larger linear acceleration than cylinder Q:
This statement is correct. As mentioned above, cylinder P has a larger linear acceleration due to its smaller radius of gyration.
c) Both cylinders reach the ground with the same translational kinetic energy:
This statement is incorrect. The translational kinetic energy of a rolling cylinder is given by the equation:
KE_trans = (1/2) * M * v^2
where M is the mass of the cylinder and v is its linear velocity. Since cylinder P reaches the ground with a larger linear velocity (due to its larger linear acceleration), it will have a greater translational kinetic energy compared to cylinder Q.
d) Cylinder Q reaches the ground with a larger angular speed:
This statement is correct. The angular speed of a rolling cylinder is given by the equation:
ω = v / R
where ω is the angular speed and R is the radius of the cylinder. Since cylinder P reaches the ground with a larger linear velocity (due to its larger linear acceleration), it will have a greater angular speed compared to cylinder Q.
Therefore, the correct statement is option D - Cylinder Q reaches the ground with a larger angular speed.
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Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer?
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Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer?.
Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?a)Both cylinders P and Q reach the ground at the same time.b)Cylinders P has larger linear acceleration than cylinder Q.c)Both cylinders reach the ground with same translational kinetic energy.d)Cylinder Q reaches the ground with larger angular speed.Correct answer is option 'D'. Can you explain this answer?.
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