The behaviour of rotating objects or objects moving along circular paths can be unintuitive because we are more used to linear motion than rotational or circular motion in our everyday lives.

Theory

Centres of Mass and Gravity

When analysing the motion or mechanical behaviour of an extended object, the extended object can be treated as it if were a point massif the average position of all of the mass in the extended object   is known. This average position of the mass is called the centre of mass, or CM. For a collection of objects whose horizontal positions of centres of mass are known, the horizontal position of the CM of the collection of objects (xCM ) can be found using:

Where mi  is the mass of object i, xi  is the horizontal position of that mass, and the Σ symbol means “add up all of the”. In words, this equation is saying that the horizontal position of the centre of mass can be found by adding up, for each mass, the product of that mass multiplied by the horizontal position of the CM of that mass, and then dividing by the total mass. Applied to a system with a few masses, this would be:

The centre of gravity, or CG,is an idea linked to the CM. If the CM is the average position of the mass in an extended object, then the gravitational force could be thought to act at this position, i.e. the CM and CG are located at the same position. This is true for objects where the acceleration due to gravity can be assumed to be roughly constant across the object. For objects smaller than about 100 m and that are within the Earth’s atmosphere, this assumption holds.

The main part of the experiment today is focused on these ideas of CG and CM, which is linked to the idea of balancing loads when loading passengers or cargo into an aircraft. Care must betaken to ensure that the centre of gravity (CG) is positioned such that it allows the safe control of an aircraft.

Circular motion

When an object is moving along a circular path, it must have a component of its acceleration which points towards the centre of the circle that the circular path follows – acentripetal acceleration (ac ). The centripetal acceleration of an object can be worked out if the tangential speed, Vt, and radial distance from the centre of the circle, T, are known using:

If an extended object is spinning or moving along a circular path, that rotational or circular motion can be considered and analysed by looking at the motion of the CM. If the CM is moving along a circular path, then there will need to be a centripetal acceleration acting on the CM to keep its motion circular. Whereas if the CM is not moving along a circular path, for example because the CM is stationary, then no centripetal acceleration is needed.

Angular momentum and precession

Angular velocity, angular momentum, and torque all have a vectorial nature. The direction of the vector can be worked out using the right hand curl rule, where the fingers curl in the rotational direction (clockwise or anti-clockwise) and the thumb gives the direction of the vector for that quantity.

Where a net torque is acting on a rotating object, that net torque causes the angular momentum to change. The vectorial natural of torque and angular momentum means that the torque and angular momentum vectors can have parallel or anti-parallel (point in opposite directions), and perpendicular components. Where the torque and angular momentum vectors have components that are parallel or anti-parallel, this causes the magnitude of the angular momentum vector to change. Where the torque and angular momentum vectors have components that are perpendicular, this causes the direction of the angular momentum vector to change.

If the magnitude of the angular momentum vector is changing, then absent any changes in the rotational inertia, the object will rotate at a faster or slower speed. Where the direction of the angular momentum vector is changing, the axis of rotation of the rotating object will itself rotate. This rotation of the angular momentum vector, and hence the axis of rotation, is called precession.

Experiment

This experiment is comprised of three parts. In part A you will find the location of a weight inside a closed tube using the CM. In part B you will investigate what you feel and see when holding a spinning disk with and without weights attached. In part C you will investigate the effect that changing the mass distribution has on rotational motion.

All pairs will start work on part A. In groups of four you will be invited by your demonstrator to attempt part B – you will have up to 25 minutes with the part B apparatus before another group would need to use it. Part C should be completed last, after both parts A and B have been completed, or while you are waiting to work on part B if you finish Part A quickly.

Part A: Finding the location of a weight inside a system using the idea of centre of mass

In this part of the experiment, you are to find the position of a weight located inside a sealed tube using the idea of the centre of mass (CM).

The apparatus you will use consists of a hollow plastic tube, with screw on/offend caps at each end. Holes drilled at each end allow a threaded metal rod to be placed inside the tube. Wingnuts and washers on the exterior of the tube at each cap hold the rodin place. Figure 1 below shows the exterior of the tube, end caps, and a short section of the rod that emerges at each end.

Figure 1: Pictures showing the system tube and end caps.

A steel weight is attached to the rod and also held in place with wingnuts and washers to prevent the weight from moving as the assembly is handled. Figure 2 on the next page shows a weight in  place on the rod.

Figure 2: Picture showing the weight on the rod pulled out of the tube for measurement.

The combination of the hollow tube, rod, wingnuts, and washers is referred to as the assembly. The position of the weight on the rod inside the tube is offset from the centre of the tube, causing the CM of the system to be shifted from its centre. An example showing a possible configuration of the weight inside the assembly is shown in Figure 3 on the next page.

Figure 3: Cross-section of the system showing the hollow tube (thick black line), metal rod (thinner black line) and weight (grey) on the metal rod inside the tube. A datum, or reference, line is also shown from which measurements of horizontal position can be made.

You can assume that the assembly, and the weight inside the assembly are both uniform in their mass distribution. However, as the weight inside the assembly is not centred, the weight distribution of the system is not balanced.

 When handling the tube ensure that you handle the tube with one hand close to each end. Do not try to handle the tube with a single hand in the middle.

To find the position of the weight inside the assembly, you’ll need to measure or workout the positions of the centres of mass of the system and the assembly. The mass of the assembly is 2052 g; The mass of the weight on the rod is 2085 g. Each system is numbered. You could use the numbered  end of the assembly as the reference or datum end for making measurements of position.

Method

1.     Make a note of the number of the system you are using on your logbook.

 When balancing the system on the ruler, take care that the system does not roll off the ruler or otherwise fall to the ground.

2.    Turn your metre rule edge on and place it on the table along the long edge. Position the system on the edge of ruler so that the system balances with neither end of the system touching the table.

3.     Place a post-it note on the top surface of the pipe at the horizontal location where the system balances on the ruler. Mark a line onto the post-it note to indicate the position where the system is balanced.

4.    Carefully remove the system from the ruler. Measure the distance from your reference or datum line to the position where the system balances. Record in your logbook.

5.     Use the ruler to measure the length of the tube, and hence find the CM of the assembly.

Part B: Investigating rotational motion with a spinning disk

In this part of the experiment, you will be investigating what you feel and see when rotating a disk that can have a mass attached. The rotational axis of each apparatus is aligned with the handles.

Method

There are four setups available:

•    Setup 1 is a disk with no mass attached,

•    Setup 2 is a disk with a mass attached close to the rotational axis,

•    Setup 3 is a disk with a mass attached further from the rotational axis,

•    Setup 4 is a disk with no mass attached, and a shackle from which the spinning disk can be suspended.

These setups are shown and indicated in Figure 4 below.

Figure 4: Pictures showing each disk setup. The number in each quadrant indicates the number of the setup, e.g. Setup 2 is shown in the top-right.

As you spin the disks ensure that:

•    the setup is held by the handles,

•    the handles are held horizontal,

•    you spin by contacting the rim of the disk.

Δ Hold the setup so that the spinning disk does not contact you or anyone around you, any loose clothing you or anyone else may be wearing, or anyone’s hair.

To stop the disk spinning, place the rim of the disk against a solid surface.

Δ If you find you cannot continue holding the setup. Stop the disk spinning, put the setup down, and rest a while.

Work through these activities as two pairs (i.e. groups of four). Each group member should try holding each disk in situations 1-6 to allow each group member to form their own opinion about what they feel. Situation 8 only needs to be observed by each group member.

1.     Using setups 1 and 2, spin the disks at roughly the same speed. Hold the disks by the handles with the handles horizontal.

2.     Note the differences in what you feel when you are holding setups 1 and 2 as the disks are spinning in your logbook.

3.     Using setups 2 and 3, spin the disks at roughly the same speed. Hold the disks by the handles with the handles horizontal.

4.     Note the differences in what you feel when you are holding setups 2 and 3 as the disks are spinning in your logbook.

5.     Using setup 3, spin it at a low rotational speed and then a higher rotational speed. For each speed, hold the disk by the handles with the handles horizontal.

6.     Note the differences in what you feel while you are holding setup 3 with different rotational speeds in your logbook.

Discussion

1.     Describe your observations from steps 1-6.

2.     Explain your observations from steps 1-6. Your explanation should discuss the effect of the

additional mass on what you feel (1-4), and address how the rotational speed of the disk affects what you feel (5-6).

3.     Using setup 4, have one person hold the setup using the handles. Set the disk spinning rapidly. Have a second person hold the shackle, and then have the first person release the handles to   allow the disk to move freely.

4.     Describe the motion of the disk in setup 4 when the disk is spinning while the apparatus is suspended using the shackle and chain and explain why the disk behaves in this way. Your explanation should clearly explain why the disk moves in one direction and not the other.

Part C: Investigating rotational motion with a changing mass distribution

In this part of the experiment, you will be investigating the effect of changing the distribution of mass of an object on the rotation of that object. You will use a ruler, some reusable and repositionable adhesive (such as Blu Tack), and two small masses. Figure 5 below gives an example showing how the masses can be attached to the ruler.

Figure 5: Picture showing metre ruler with masses attached using adhesive. The ruler should be gripped at its mid-point, i.e. at the 50 cm/500 mm mark.

Method

1.     Hold the metre ruler between your fingers and thumb at the 50 cm/500 mm mark. Try rotating the ruler back and forth.

2.     Now use the adhesive to securely attach the masses at the 40 cm/400 mm mark and the

60 cm/600 mm mark. Try rotating the ruler back and forth, holding the ruler at the 50 cm/500 mm mark.

3.     Now move the masses to the 5 cm/50 mm mark and the 95 cm/950 mm mark. Try rotating the ruler back and forth, holding the ruler at the 50 cm/500 mm mark.

Discussion

1.     Discuss what you felt as you tried to rotate the ruler with attached masses back and forth in steps 2 and 3. Was one situation easier than the other? Explain your reasoning using your knowledge of physics.