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Chapter 5: Problem 595

A Pulley of radius \(2 \mathrm{~m}\) is rotated about its axis by a force \(F=\left(20 t-5 t^{2}\right) N\) where \(t\) is in sec applied tangentially. If the moment of inertia of the Pulley about its axis of rotation is $10 \mathrm{KgM}^{2}$, the number of rotations made by the pulley before its direction of motion is reversed is : \(\\{\mathrm{A}\\}\) more than 3 but less then 6 \(\\{\mathrm{B}\\}\) more than 6 but less then 9 \(\\{\mathrm{C}\\}\) more than 9 \\{D \\} Less then 3

Short Answer

Expert verified
The number of rotations made by the pulley before its direction of motion is reversed is more than 9 rotations (Option C).

Step by step solution

01

Calculate the torque on the pulley

In order to find the torque on the pulley, we need to multiply the force applied by the radius of the pulley. The force applied on the pulley is given by \(F(t) = (20t - 5t^2) N\). Multiply this force by the radius of the pulley, which is 2 meters: \[τ(t) = r \cdot F(t) = 2 \cdot (20t - 5t^2) = 40t - 10t^2 \mathrm{~N m}\]
02

Calculate the angular acceleration of the pulley

Now that we have the torque on the pulley, we can use the relationship between torque and moment of inertia, \(\tau = I \cdot \alpha\), where \(\tau\) is the torque, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration. Solve for \(\alpha\): \[ \alpha(t) = \frac{\tau(t)}{I} = \frac{40t - 10t^2}{10} = 4t - t^2 \mathrm{~rad/s^2}\]
03

Integrate the angular acceleration function

Next, integrate the angular acceleration function with respect to time to find the angular velocity function, \(\omega(t)\). Don't forget to add an integration constant, \(C_1\): \[ \omega(t) = \int \alpha(t) dt = \int (4t - t^2)dt = 2t^2 - \frac{1}{3}t^3 + C_1 \mathrm{~rad/s}\]
04

Determine when the angular velocity becomes zero

To find when the direction of motion is reversed, we need to find when the angular velocity becomes zero. Solving for \(t\) in \(\omega(t) = 0\): \[ 2t^2 - \frac{1}{3}t^3 + C_1 = 0\] Assuming the pulley starts from rest, \(C_1 = 0\). So, solve the equation for \(t\): \[ 2t^2 - \frac{1}{3}t^3 = 0\] \[\frac{1}{3}t^3 = 2t^2\] \[ t = \frac{6}{1}\] So, the pulley reverses its direction at \(t = 6s\).
05

Calculate the total number of rotations before reversal

Finally, to find the total number of rotations, we need to find the total angle covered by the pulley by integrating the angular velocity function: \[θ(t) = \int \omega(t) dt = \int_0^6 (2t^2 - \frac{1}{3}t^3) dt = \left[\frac{2}{3}t^3 - \frac{1}{12}t^4\right]_0^6\] \[ θ(6) = \frac{2}{3}(6)^3 - \frac{1}{12}(6)^4 = 72\pi\] Now, divide this angle by a full rotation, which is \(2\pi\) radians: \[ \text{number of rotations} = \frac{72\pi}{2\pi} = 36\] The number of rotations made by the pulley before its direction of motion is reversed is 36. Which corresponds to option C: More than 9 rotations.

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Most popular questions from this chapter

A uniform rod of length \(\mathrm{L}\) is suspended from one end such that it is free to rotate about an axis passing through that end and perpendicular to the length, what maximum speed must be imparted to the lower end so that the rod completes one full revolution? \(\\{\mathrm{A}\\} \sqrt{(2 \mathrm{~g} \mathrm{~L})}\) \(\\{\mathrm{B}\\} 2 \sqrt{(\mathrm{gL})}\) \(\\{C\\} \sqrt{(6 g L)}\) \(\\{\mathrm{D}\\} 2 \sqrt{(2 g \mathrm{~L})}\)

A rod of length L rotate about an axis passing through its centre and normal to its length with an angular velocity \(\omega\). If A is the cross-section and \(D\) is the density of material of rod. Find its rotational $\mathrm{K} . \mathrm{E}$. \(\\{\mathrm{A}\\}(1 / 2) \mathrm{AL}^{3} \mathrm{D} \omega^{2}\) \\{B \(\\}(1 / 6) \mathrm{AL}^{3} \mathrm{D} \omega^{2}\) \(\\{C\\}(1 / 24) A L^{3} D \omega^{2}\) \(\\{\mathrm{D}\\}(1 / 12) \mathrm{AL}^{3} \mathrm{D} \omega^{2}\)

A cylinder of mass \(5 \mathrm{~kg}\) and radius \(30 \mathrm{~cm}\), and free to rotate about its axis, receives an angular impulse of $3 \mathrm{~kg} \mathrm{M}^{2} \mathrm{~S}^{-1}$ initially followed by a similar impulse after every \(4 \mathrm{sec}\). what is the angular speed of the cylinder 30 sec after initial impulse? The cylinder is at rest initially. \(\\{\mathrm{A}\\} 106.7 \mathrm{rad} \mathrm{S}^{-1}\) \\{B\\} \(206.7 \mathrm{rad} \mathrm{S}^{-1}\) \\{C\\} \(107.6 \mathrm{rad} \mathrm{S}^{-1}\) \\{D \(\\} 207.6 \mathrm{rad} \mathrm{S}^{-1}\)

Two disc of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio \(1: 3\). The moment of inertia of these disc about the respective axes passing through their centres and perpendicular to their planes will be in the ratio. \(\\{\mathrm{A}\\} 1: 3\) \\{B\\} \(3: 1\) \\{C\\} \(1: 9\) \(\\{\mathrm{D}\\} 9: 1\)

A small object of uniform density rolls up a curved surface with initial velocity 'u'. It reaches up to maximum height of $3 \mathrm{v}^{2} / 4 \mathrm{~g}$ with respect to initial position then the object is \(\\{\mathrm{A}\\}\) ring \(\\{B\\}\) solid sphere \(\\{\mathrm{C}\\}\) disc \\{D\\} hollow sphere
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