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Question #42403

Difficulty: 8/10 | 🟢 Free

The Question

A uniform cylindrical roll of tape of initial mass $M_0$ and initial radius $R_0$ is mounted on a fixed, frictionless horizontal axle. The free end of the tape is pulled with a constant horizontal force $F$ . As the tape unwinds, it does not slip or stretch, and its mass per unit length is $\lambda$ . The roll can be modeled as a uniform solid cylinder at all times. Assuming the unwound portion of the tape moves horizontally without sagging, which of the following expressions gives the angular velocity $\omega$ of the roll when its radius has decreased to $r$ ?

Options

A.
$\frac{2}{r} \sqrt{ \frac{F(R_0^2 - r^2)}{\lambda(2 R_0^2 - r^2)} }$
B.
$\frac{2}{r^2} \sqrt{ \frac{F(R_0^2 - r^2)}{\lambda} }$
C.
$\frac{2}{r} \sqrt{ \frac{F(R_0^2 - r^2)}{\lambda R_0^2} }$
D.
$\frac{2}{r} \sqrt{ \frac{F(R_0^2 - r^2)}{\lambda(R_0^2 + r^2)} }$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Angular Velocity vs Linear Velocity|Pure Rolling Condition|Integration

Why This Question Is Tricky

Common Mistakes

The primary trap is applying $\tau = I \alpha$ with $\tau = Fr$ , incorrectly assuming the tension in the tape at the detachment point equals the applied force $F$ .

Because the unwound tape has mass and is accelerating, $T \neq F$ .

Furthermore, students often misapply $dL/dt = \tau_{ext}$ by writing $d(I\omega)/dt = Fr$ , ignoring both the angular momentum flux of the shedding mass and the kinetic energy of the unwound tape.

The Work-Energy theorem elegantly bypasses the complex variable-mass non-inertial dynamics, provided the kinetic energy of the unwound tape is correctly included.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Angular Velocity vs Linear Velocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free