A solid uniform sphere of mass 1.85 kg and diameter 45.0 cm spins about an axle through its center.Starting with an angular velocity of 2.40 rev/s,it stops after turning through 18.2 rev with uniform acceleration.The net torque acting on this sphere as it is slowing down is closest to
A) 0.00593 N • m.
B) 0.0372 N • m.
C) 0.0466 N • m.
D) 0.0620 N • m.
E) 0.149 N • m.

Question

Quizplus · Accepted Answer

To find the net torque, we first need to calculate the angular acceleration. The angular displacement ($\theta$) is given in revolutions (18.2 rev), which we convert to radians by multiplying by $2\pi$ (since 1 rev = $2\pi$ radians). Thus, $\theta = 18.2 \times 2\pi$ radians. The initial angular velocity ($\omega_0$) is 2.40 rev/s, which also needs to be converted to radians per second, giving $\omega_0 = 2.40 \times 2\pi$ rad/s. The final angular velocity ($\omega$) is 0 rad/s since the sphere stops.The angular acceleration ($\alpha$) can be found using the equation $\omega^2 = \omega_0^2 + 2\alpha\theta$, solving for $\alpha$ gives $\alpha = (\omega^2 - \omega_0^2) / (2\theta)$.Substituting the given values and solving for $\alpha$ yields a negative value (since the sphere is decelerating), but we're interested in the magnitude for calculating torque.The moment of inertia ($I$) of a solid sphere about an axis through its center is given by $I = 2/5 MR^2$, where $M$ is the mass and $R$ is the radius. The radius $R$ is half the diameter, so $R = 45.0 \, \text{cm} / 2 = 22.5 \, \text{cm} = 0.225 \, \text{m}$.Substituting $M = 1.85 \, \text{kg}$ and $R = 0.225 \, \text{m}$ into the formula for $I$ gives the moment of inertia.Finally, the torque ($\tau$) is related to the angular acceleration and the moment of inertia by $\tau = I\alpha$. Substituting the values of $I$ and $\alpha$ into this equation gives the net torque.After performing these calculations, the result matches closest to the value given in option B, 0.0372 N • m. This involves calculating the angular acceleration from the given conditions, determining the moment of inertia of the sphere, and then using these to find the torque.

A Solid Uniform Sphere of Mass 1