Mobiles & Static Equilibrium

Alexander Calder Image by Digital Image, Corbis

Reading: Knight, Physics for Scientists and Engineers, Chapter 12, section 8. Introduction: We divide physics, more specifically mechanics, into two main areas: Kinematics, which de-scribes motion and Dynamics, which explains motion. Today we study a special case of dynam-ics, Statics, which explains why things sometimes don't move!

In order for a system of objects to remain stationary, or static, we require three things. First, we must be able to find an inertial frame of reference for which all component objects have zero lin-ear velocity and zero angular velocity. Second the net force on each object within the system must be zero, so that these zero linear velocities remain zero. Finally, the net torque on each ob-ject must also be zero, so that it does not rotate. This last requirement is sometimes the trickiest to implement in everyday life: for example, when an object tips and falls over, its motion is pre-dominated rotational (like a ladder falling away from a wall).

Here is a simple (and extremely boring) example: put a thick book on a flat horizontal table. In this situation, the normal force automatically adjusts to exactly balance the gravitational force perpendicular to the table's surface, so the net force on the book is zero. The book also experi-ences a zero net torque: If we measure torques with respect to the book's center of mass, then the

gravitational force exerts no torque, since it acts at the center of mass. Meanwhile, the normal force from the table is evenly distributed along the lower surface of the book, so it also exerts no net torque. The book starts at rest and does not accelerate; it is in static equilibrium. Now let’s make things a little more interesting: imagine tilting the table slightly, and assume that static friction is sufficient to prevent sliding. The book again neither translates nor rotates. The gravitational force still acts through the center of mass, and so does not exert a torque about that point. However, the force of static friction does exert a torque on the book, since it acts at the lower surface of the book, which is some distance removed from its center of mass. So why doesn’t the book rotate? What other torque acts on the book? This is a subtle issue; think about how the normal force is distributed along the bottom surface of the book... In today’s lab, we will study the conditions that describe the static equilibrium of a mobile, a se-ries of hanging masses connected with strings. Unlike most artists’ mobiles, however, ours isn’t just a series of sticks: it contains a two-dimensional sheet! Goals: Understand static equilibrium, in which a system of objects remains stationary. Become proficient at calculating and visualizing torques along several axes. Note the role of center of mass in simplifying calculations of equilibrium. Equipment: One metal mesh square One set of masses Mass hangers Beaded hooks Half-meter stick Post, stand and associated hardware Lab Activity: Unlike a normal mobile, which consists of masses hung from a set of interlinked one-dimensional bars, we will construct a mobile wherein the “bar” is actually a two-dimensional metal mesh sheet. Additionally, this sheet will be hung not from its center of mass, but from a point offset from it. First, set up the post and stand to accommo-date a single string from which will hang the metal mesh square. The point of attachment of the string to the mesh should be offset from the center of the sheet- you can choose the precise position for this point of attachment. Next, hang two or three masses from the sheet, at locations also of your choosing, such that the overall structure forms a static equilibrium with the metal mesh square roughly horizontal. There are two ways that you can reach this goal: (1) Design and Build: use the equations of static equilibrium to design a particular arrangement of masses that should be in static equilibrium and then build the resulting design. (2) Build and Test: construct a balanced two-dimensional mobile by trial

and error, and once it is working well, measure its precise geometry and verify mathematically that it satisfies the equations of static equilibrium. Either strategy is fine, so long as at the end you have demonstrated experimentally that your design works, and demonstrated mathematically why it works. There are several ways in which you can attach the masses to the metal mesh square. With the beaded hooks, you can thread them through the metal mesh and hand a mass hanger from it, or you can place the mass directly on the metal mesh square and use the beaded hooks to pin them in place. The second method requires that you take care to locate the proper positions for the masses before pinning them to the metal mesh square so that it doesn’t tip and your weights fall off. Make sure the paperclips are attached firmly and don’t slip when you rock the mesh square. In order for your mobile to prefer an orientation where the mesh square is horizontal (i.e. parallel to the ground), it is important that the masses hung from it be heavy enough to exert torques on the sheet that are large enough to just slightly flex the sheet (you don’t need much flexing- just a little is enough). Otherwise, the sheet will be in what is called a neutral equilibrium: equally sta-ble (with no net force or torque) over a wide range in angles. Since our metal mesh square is free to rotate about both the x and y axes, the torque condition

for the net torque on the mesh square includes both x and y components. In calculating torques, you can assume that the gravitational force acts on the center of mass of each object. Apply the conditions of static equilibrium to each section of the mobile in turn, starting from the simplest cases (i.e. the masses at the bottom, which are connected to only one string). Don't for-get to take into account the mass of the mesh square! You may assume that, once the system is in static equilibrium, the tensions in the strings have adjusted themselves so as to yield zero net force on each component part. To hand in: Draw a diagram of your final setup (hand drawn is fine), indicating all the relevant forces and torques. Draw a free body diagram for each mass and the mesh square, and demonstrate that the forces are in equilibrium (how do you know?). Demonstrate that your mobile is also in static rotational equilibrium via detailed calculations of the net torque on the mesh square. Be sure to define your coordinate system (where is the origin, and x and y axes?). Are each of the hanging masses in static rotational equilibrium? How do you know? Since you cannot measure the di-mensions and masses of your mobile with perfect accuracy, you will have some slight error in your final results. Discuss whether the discrepancies between the observed behavior of the mo-bile and your calculations of static equilibrium are consistent with your estimates of the errors in your measurements. Extra Credit Explain precisely why it is important, if we want the mobile to be in equilibrium only when the metal square is oriented horizontal (and not when skewed at some angle), that the hanging masses be large enough to flex the square.