Using the “Clicker” If you have a clicker now, and did not do this last time, please enter your ID in your clicker. First, turn on your clicker by sliding the power switch, on the left, up. Next, store your student number in the clicker. You only have to do this once. Press the * button to enter the setup menu. Press the up arrow button to get to ID Press the big green arrow key Press the T button, then the up arrow to get a U Enter the rest of your BU ID. Press the big green arrow key.

Torque Forces can produce torques. The magnitude of a torque depends on the force, the direction of the force, and where the force is applied. The magnitude of the torque is . is measured from the axis of rotation to the line of the force, and is the angle between and . To find the direction of a torque from a force, pin the object at the axis of rotation and push on it with the force. We can say that the torque from that force is whichever direction the object spins (counterclockwise, in the picture above). Torque is zero when and are along the same line. Torque is maximum when and are perpendicular.

Equilibrium For an object to remain in equilibrium, two conditions must be met. The object must have no net force: and no net torque:

Worksheet, part 1 A uniform rod with a length L and a mass m is attached to a wall by a hinge at the left end. A string will hold the rod in a horizontal position; the string can be tied to one of three points, lettered A-C, on the rod. The other end of the string can be tied to one of three hooks, numbered 1-3, above the rod. This system could be a simple model of a broken arm you want to immobilize with a sling.

Sling, part 1 How would you attach a string so the rod is held in a horizontal position but the hinge exerts no force at all on the rod? A  1. A  2. A  3. B  1 or B  3. B  2. C  1. C  2. C  3. It can’t be done.

Sling, part 2 How would you attach a string so the rod is held in a horizontal position while the force exerted on the rod by the hinge has no horizontal component, but has a non-zero vertical component directed straight up? A  1. A  2. A  3. B  1 or B  3. B  2. C  1. C  2. C  3. It can’t be done.

Sling, part 3 How would you attach a string so the rod is held in a horizontal position while the force exerted on the rod by the hinge has no vertical component, but has a non-zero horizontal component? A  1. A  2. A  3. B  1 or B  3. B  2. C  1. C  2. C  3. It can’t be done.

A balanced beam A uniform beam sits on two identical scales. Scale A is farther from the center than scale B, but the beam remains in equilibrium. Which scale shows a higher reading? How far to the left could scale B be moved without the beam tipping over?

A balanced beam A uniform beam sits on two identical scales. Scale A is farther from the center than scale B, but the beam remains in equilibrium. Which scale shows a higher reading? Scale B – it is closer to the center of gravity. How far to the left could scale B be moved without the beam tipping over?

A balanced beam A uniform beam sits on two identical scales. Scale A is farther from the center than scale B, but the beam remains in equilibrium. Which scale shows a higher reading? Scale B – it is closer to the center of gravity. How far to the left could scale B be moved without the beam tipping over? To the center of the beam, but no farther. The beam’s center of gravity must be between the supports to be stable.

A balanced beam Could you place a weight on the beam without the reading on scale A changing? Could you place a weight on the beam and cause the reading on scale A to decrease?

A balanced beam Could you place a weight on the beam without the reading on scale A changing? Yes – place it directly over scale B. Could you place a weight on the beam and cause the reading on scale A to decrease?

A balanced beam Could you place a weight on the beam without the reading on scale A changing? Yes – place it directly over scale B. Could you place a weight on the beam and cause the reading on scale A to decrease? Yes – put it on the beam to the right of scale B.

A balanced beam What happens to the scale readings when scale B is moved to the right? 1. A is unchanged, B goes up. 2. A is unchanged, B goes down. 3. Both readings decrease. 4. Both readings increase. 5. A goes up, B goes down. 6. A goes down, B goes up. 7. None of the above.

The human spine Equilibrium ideas can be applied to the human body, including the spine. If you bend your upper body so it is horizontal, you put a lot of stress on the lumbrosacral disk, the disk separating the lowest vertebra from the tailbone (the sacrum). Picking something up is even worse. Simulation Treat the spine as a pivoted bar. There are essentially three forces acting on this bar: The force of gravity, mg, acting on the upper body (this is about 65% of the body weight). The tension in the back muscles. This can be considered as one force T that acts at an angle of about 12° to the horizontal when the upper body is horizontal. The support force F from the tailbone, which also acts at a small angle measured from the horizontal.

The human spine At equilibrium, take torques about the tailbone. T is applied about 10% farther from the tailbone than the force of gravity. The components of the support force F can be found by summing forces. For a person with a weight of 600 N, the upper body weighs almost 400 N, while T and F are around 1700 N. Picking up something with your arms that has a weight of 100 N increases both T and F by about 600 N! The moral of the story: bend your legs instead of your back. Picking up a 100 N bag of groceries by bending at the knee produces a force of about 500 N on the bottom disk in the spine. The force is 4-5 times larger if you bend your back!

Center of gravity The center-of-gravity of an object is the point that moves as though the weight of the object is concentrated there. The center-of-gravity is given by: In a uniform gravitational field, the center-of-mass and the center-of-gravity are the same point. They're different if the gravitational field is non-uniform. The center-of-gravity can be found by hanging an object from a support. In stable equilibrium, the center-of-gravity is directly below the support.

Red and blue rods Two rods of the same shape are held at their centers and rotated back and forth. The red one is much easier to rotate than the blue one. What is the best possible explanation for this? 1. The red one has more mass. 2. The blue one has more mass. 3. The red one has its mass concentrated more toward the center; the blue one has its mass concentrated more toward the ends. 4. The blue one has its mass concentrated more toward the center; the red one has its mass concentrated more toward the ends. 5. Either 1 or 3 6. Either 1 or 4 7. Either 2 or 3 8. Either 2 or 4 9. Due to the nature of light, red objects are just inherently easier to spin than blue objects are.

Newton’s First Law for Rotation An object at rest tends to remain at rest, and an object that is spinning tends to spin with a constant angular velocity, unless it is acted on by a nonzero net torque or there is a change in the way the object's mass is distributed. The net torque is the vector sum of all the torques acting on an object. The tendency of an object to maintain its state of motion is known as inertia. For straight-line motion mass is the measure of inertia, but mass by itself is not enough to define rotational inertia.

Rotational Inertia How hard it is to get something to spin, or to change an object's rate of spin, depends on the mass, and on how the mass is distributed relative to the axis of rotation. Rotational inertia, or moment of inertia, accounts for all these factors. The moment of inertia, I, is the rotational equivalent of mass. For an object like a ball on a string, where all the mass is the same distance away from the axis of rotation: If the mass is distributed at different distances from the rotation axis, the moment of inertia can be hard to calculate. It's much easier to look up expressions for I from the table on page 291 in the book (page 10-15 in Essential Physics).

A table of rotational inertias

The parallel axis theorem If you know the rotational inertia of an object of mass m when it rotates about an axis that passes through its center of mass, the object’s rotational inertia when it rotates about a parallel axis a distance h away is:

Worksheet, part 2 When a system is made up of several objects, its total rotational inertia about a particular axis is the sum of the rotational inertias of the individual objects for rotation about that axis. What is the system’s rotational inertia in the first case? Each block has a mass of m/3, and the rod, of negligible mass, has a length L.

Worksheet, part 2 When a system is made up of several objects, its total rotational inertia about a particular axis is the sum of the rotational inertias of the individual objects for rotation about that axis. What is the system’s rotational inertia in the first case?

Worksheet, part 2 In the second case, do we expect the rotational inertia to be larger, smaller, or the same as the rotational inertia in the first case? What is the system’s rotational inertia in the second case?

Worksheet, part 2 In the second case, do we expect the rotational inertia to be larger, smaller, or the same as the rotational inertia in the first case? Larger – the mass is farther from the axis. What is the system’s rotational inertia in the second case?

Worksheet, part 2 In the second case, do we expect the rotational inertia to be larger, smaller, or the same as the rotational inertia in the first case? Larger – the mass is farther from the axis. What is the system’s rotational inertia in the second case? The parallel-axis theorem gives the same result.

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