Question

A disk with mass m = 6.1 kg and radius R = 0.39 m hangs from a rope attached to the ceiling. The disk spins on its axis at a distance r = 1.51 m from the rope and at a frequency f = 19.8 rev/s (with a direction shown by the arrow).

1)What is the magnitude of the angular momentum of the spinning disk?

2)What is the torque due to gravity on the disk? if found this to be --> 90.2678 N-m

3)What is the period of precession for this gyroscope?

4)What is the direction of the angular momentum of the spinning disk at the instant shown in the picture?

up
down
left
right

5)What is the direction of the precession of the gyroscope?

it does not precess
clockwise as seen from above (looking down the rope)
counterclockwise as seen from above (looking down the rope)

Answer 1

Concepts and reason

The concepts used to solve this problem are the rotational angular momentum, torque due to gravity on the disk and right-hand thumb rule are used here.

First find out the value of moment of inertia of spinning disk and angular frequency of the disk and substitute both the values in the expression of angular momentum. Using the right-hand rule, find out the direction of angular momentum.

In the next part, find the torque due to gravity on the disk by multiplying the gravity force with the distance.

In the next part, use the expression of rate of precession and from that value find the period of precession.

At last, use right hand rule to find the direction of precession of gyroscope.

Fundamentals

The expression of the magnitude of the rotational angular momentum of the spinning disk is given as follows:

$L = I\omega$

Here, I is the moment of inertial and $\omega$is the angular speed for one complete rotation.

The direction of angular momentum is given using the Right-hand rule. Curl the fingers in the direction of rotation then the thumb of your right hand will give the direction of angular momentum.

The expression of angular speed is given as follows:

$\omega = 2\pi f$

Here, f is the frequency of rotation.

The expression of the rate of precession of the spinning disk is given as follows:

${\rm{Rate of precession}} = \frac{{{\tau _{gravity}}}}{L}$

The expression of the period of precession is given as follows:

$T = \frac{{2\pi }}{{{\rm{rate of precession}}}}$

(1)

The moment of inertia of the spinning disk is given as follows:

$I = \frac{1}{2}m{R^2}$

The angular frequency of the spinning disk is given as follows:

$\omega = 2\pi f$

Use the expression $I = \frac{1}{2}m{R^2}$and $\omega = 2\pi f$in the expression $L = I\omega$.

$\begin{array}{c}\\L = \left( {\frac{1}{2}m{R^2}} \right)\left( {2\pi f} \right)\\\\ = m{R^2}\pi f\\\end{array}$

Substitute 6.1 kg for m, 0.39 m for R and $19.8{\rm{ rev/s}}$for $f$in the above equation of angular momentum.

$\begin{array}{c}\\L = \left( {6.1{\rm{ kg}}} \right){\left( {0.39{\rm{ m}}} \right)^2}\left( {3.14} \right)\left( {19.8{\rm{ rev/s}}} \right)\\\\ = 57.7{\rm{ kg}} \cdot {{\rm{m}}^2}/{\rm{s}}\\\end{array}$

(2)

The expression of the torque due to gravity on the disk is given as follows:

${\tau _{gravity}} = mg\left( r \right)$

Here, mg is the force acting on disk due to gravity and r is the distance of the disk from the rope.

Substitute 6.1 kg fir m, $9.8{\rm{ m/}}{{\rm{s}}^2}$for g and 1.51 m for r in the expression ${\tau _{gravity}} = mg\left( r \right)$.

$\begin{array}{c}\\{\tau _{gravity}} = \left( {6.1{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^2}} \right)\left( {1.51{\rm{ m}}} \right)\\\\ = 90.27{\rm{ N}} \cdot {\rm{m}}\\\end{array}$

(3)

Substitute $90.27{\rm{ N}} \cdot {\rm{m}}$for ${\tau _{gravity}}$and $57.7{\rm{ kg}} \cdot {{\rm{m}}^2}/{\rm{s}}$for L in the expression ${\rm{Rate of precession}} = \frac{{{\tau _{gravity}}}}{L}$.

$\begin{array}{c}\\{\rm{Rate of precession}} = \frac{{90.27{\rm{ N}} \cdot {\rm{m}}}}{{57.7{\rm{ kg}} \cdot {{\rm{m}}^2}/{\rm{s}}}}\\\\ = 1.56{\rm{ /s}}\\\end{array}$

Substitute 1.56 /s for Rate of precession in the expression$T = \frac{{2\pi }}{{{\rm{rate of precession}}}}$.

$\begin{array}{c}\\T = \frac{{2\pi }}{{1.56{\rm{ /s}}}}\\\\ = 4.02{\rm{ s}}\\\end{array}$

(4)

The direction of the angular momentum is evaluated using the right-hand rule. Curl your finder in the direction of $\omega$, thumb points in the direction of angular momentum.

Here, if you curl your fingers in the anticlockwise direction, thumb points towards right.

Hence, the direction of angular momentum is towards right.

(5)

The direction of precession is evaluated using the direction of angular acceleration. First, find the direction of torque, place the fingers in the direction of r, curl them in the direction of F, and thumb points in the direction of torque.

The toque and angular acceleration are in the same direction (i.e. into the plane). Since, the disk is attached with the rope. Therefore, the direction of the precession is towards clockwise as seen from above.

Ans: Part 1

The magnitude of the angular momentum of the spinning disk is $57.7{\rm{ kg}} \cdot {{\rm{m}}^2}/{\rm{s}}$.

Part 2

The magnitude of torque due to gravity on disk is $90.27{\rm{ N}} \cdot {\rm{m}}$.

Part 3

The period of precession of the gyroscope is 4.02 s.

Part 4

The direction of angular momentum of the spinning disk is towards right.

Part 5

The direction of precession of the gyroscope is towards clockwise direction.

Answer 2

1) Find the moment of Inertia I = 1/2m*R^2 = .463905

Now to fined Angular momentum (L) = I*f *2pi = 57.71 (kg*m^2/s)

2) Find torque by taking t= m*g*r t = (6.1)*(9.8)*(1.51) = 90.2678 (nm)

3) Now this on is slightly more tricky

First Angular Frequency so (Omega = (2gd) / r^{2} w) or t/L = (90.2678)/(57.71) = 1.564 now to find period justtake (2pi)/(1.564) = 4.0153(s)

4) Right (Use righthand rule)

5) Clockwise (also can use righthand rule)

I think these are right I had different numbers on the one I did but it will give you a idea of what to do even if they are wrong.

Answer 3

1. L = I*w I = 1/2mR^2 w=18 * 2pi 2. m * 9.81 * radius of axis(1.51) 3. procession rate = torque/angular momentum (answer 2/ answer 3) 2pi/procession rate= period of procession 4. right 5. clockwise