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title: "A uniform solid sphere of mass \\(M\\), radius \\(R\\), and rotational inertia \\(I = \\dfrac{2}{5}MR^2\\) is placed on a flat horizontal surface and is initially at rest. Starting at time \\(t = 0\\), a time-dependent frictional force of magnitude \\(f(t) = F_0 e^{-bt}\\) acts tangentially at the surface of the sphere, where \\(F_0\\) and \\(b\\) are positive constants. Which of the following expressions correctly represents the angular velocity \\(\\omega(t)\\) of the sphere as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/117561/"
date_modified: "2026-08-04T07:49:06+00:00"
---

# A uniform solid sphere of mass \(M\), radius \(R\), and rotational inertia \(I = \dfrac{2}{5}MR^2\) is placed on a flat horizontal surface and is initially at rest. Starting at time \(t = 0\), a time-dependent frictional force of magnitude \(f(t) = F_0 e^{-bt}\) acts tangentially at the surface of the sphere, where \(F_0\) and \(b\) are positive constants. Which of the following expressions correctly represents the angular velocity \(\omega(t)\) of the sphere as a function of time \(t\)?

A uniform solid sphere of mass \(M\), radius \(R\), and rotational inertia \(I = \dfrac{2}{5}MR^2\) is placed on a flat horizontal surface and is initially at rest. Starting at time \(t = 0\), a time-dependent frictional force of magnitude \(f(t) = F_0 e^{-bt}\) acts tangentially at the surface of the sphere, where \(F_0\) and \(b\) are positive constants. Which of the following expressions correctly represents the angular velocity \(\omega(t)\) of the sphere as a function of time \(t\)?

- **A.** \(\omega(t) = \dfrac{2 F_0}{5 b M R} \left( 1 - e^{-bt} \right)\)
- **B.** \(\omega(t) = \dfrac{5 F_0 b}{2 M R} \left( 1 - e^{-bt} \right)\)
- **C.** \(\omega(t) = \dfrac{5 F_0}{2 b M R} \left( 1 - e^{-bt} \right)\)
- **D.** \(\omega(t) = \dfrac{5 F_0}{2 b M R} e^{-bt}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117561/*