Physics - Chapter 8: Rotational Motion

centripetal acceleration

acceleration toward the center of a curved or circular path

centrifugal force

an apparent outward force on a rotating or revolving body an apparent force that acts outward on a body moving around a center, arising from the body's inertia. Centrifugal means "center-fleeing" or "away from the center."

The centrifugal force effect is caused not by a real force, but

by inertia—the tendency of the moving object to follow a straight-line path.

Centrifugal force is not part of an interaction, but it is a

consequence of rotation. It is therefore called an apparent force, or a fictitious force.

tangential speed is ____ __________to distance from the axis for any given rotational speed

directly proportional

lever arm

distance from axis of rotation to the spot where force is applied

linear speed

distance traveled per unit of time

we have the greatest tangential speed

farthest from Earth's axis, at the equator. Centrifugal force is therefore greatest for us when we are at the equator and zero for us at the poles, where we have no tangential speed.

Whenever a rotating body contracts,

its rotational speed increases. Man on rotating platform holding weights: When he pulls the weights inward, the rotational inertia of his body and the weights is considerably reduced. What is the result? His rotational speed increases!

A point on the outside edge of a merry-go-round or turntable travels a greater distance in one complete rotation than does a point nearer the center. Traveling a greater distance in the same time means a greater ____.

speed

tangential acceleration

when tangential speed undergoes change; Any change in tangential speed indicates an acceleration parallel to tangential motion.

Just for fun, and since we're discussing round things, why are manhole covers circular in shape?

A square or any other shape can be tilted sideways and can fall in the hole. A round cover wont fall in the hole.

Where is the CG of a donut?

In the center of the hole!

Can an object have more than one CG?

No. A rigid object has one CG. If an object is nonrigid, such as a piece of clay or putty, and is distorted into different shapes, then its CG may change as its shape changes. Even then, it has one CG for any given shape.

rotational inertia

Reluctance or apparent resistance of an object to change its state of rotation, determined by the distribution of the mass of the object and the location of the axis of rotation or revolution. Simply put: the property of an object to resist changes in its rotational state of motion.

Consider the balanced seesaw in Figure 8.18. Suppose the girl on the left suddenly gains 50 N, such as by being handed a bag of apples. Where should she sit in order to be in balance, assuming the heavier boy does not move?

She should sit ½ m closer to the center. Then her lever arm is 2.5 m. This checks: 300 N × 2.5 m = 500 N × 1.5 m.

Trains ride on a pair of tracks. For straight-line motion, both tracks are the same length. Not so for tracks along a curve, however. Which track is longer: the one on the outside of the curve or the one on the inside?

Similar to Figure 8.1a, the track on the outside of the curve is longer—just as the circumference of a circle of greater radius is longer.

Imagine a ladybug sitting halfway between the rotational axis and the outer edge of the turntable in Figure 8.1b. When the turntable has a rotational speed of 20 RPM and the bug has a tangential speed of 2 cm/s, what will be the rotational and tangential speeds of her friend who sits at the outer edge?

Since all parts of the turntable have the same rotational speed, her friend also rotates at 20 RPM. Tangential speed is a different story: Since the friend is twice as far from the axis of rotation, she moves twice as fast—4 cm/s.

Consider balancing a hammer upright on the tip of your finger. The head is likely heavier than the handle. Is it easier to balance with the end of the handle on your fingertip, with the head at the top, or the other way around?

Stand the hammer with the handle on your fingertip and the head at the top. Why? Because it will have greater rotational inertia this way and be more resistant to a rotational change. (Try this yourself by balancing a spoon both ways on your fingertip.) Acrobats who balance a long vertical pole have an easier task when their friends are at the top of the pole. A pole empty at the top has less rotational inertia and is more difficult to balance!

A uniform meterstick supported at the 25-cm mark balances when a 1-kg rock is suspended at the 0-cm end. What is the mass of the meterstick?

The mass of the meterstick is 1 kg. Why? The system is in equilibrium, so any torques must be balanced: The torque produced by the weight of the rock is balanced by the equal but oppositely directed torque produced by the weight of the stick applied at its CG, the 50-cm mark. The support force at the 25-cm mark is applied midway between the rock and the stick's CG, so the lever arms about the support point are equal (25 cm). This means that the weights (and hence the masses) of the rock and stick are also equal. (Note that we don't have to go through the laborious task of considering the fractional parts of the stick's weight on either side of the fulcrum because the CG of the whole stick really is at one point—the 50-cm mark!) Interestingly, the CG of the rock + stick system is at the 25-cm mark—directly above the fulcrum.

A heavy iron ball is attached by a spring to a rotating platform, as shown in the sketch. Two observers, one in the rotating frame and one on the ground at rest, observe its motion. Which observer sees the ball being pulled outward, stretching the spring? Which sees the spring pulling the ball into circular motion?

The observer in the reference frame of the rotating platform states that a centrifugal force pulls radially outward on the ball, which stretches the spring. The observer in the rest frame states that a centripetal force supplied by the stretched spring pulls the ball into circular motion. Only the observer in the rest frame can identify an action-reaction pair of forces; where action is spring on ball, reaction is ball on spring. The rotating observer can't identify a reaction counterpart to the centrifugal force because there isn't any!

If a pipe effectively extends a wrench handle to three times its length, by how much will the torque increase for the same applied force?

Three times more leverage for the same force produces three times more torque. (Caution: This method of increasing torque sometimes results in shearing off the bolt!)

Consider a pair of metersticks standing nearly upright against a wall. If you release them, they'll rotate to the floor at the same time. But what if one meterstick has a massive hunk of clay stuck to its top end? Will it rotate to the floor in a longer or shorter time?

Try it and see! (If you don't have clay, fashion something equivalent.)

conservation of angular momentum

When no external torque acts on an object or a system of objects, no change of angular momentum can occur. If no external net torque acts on a rotating system, the angular momentum of that system remains constant.

If Earth were to spin faster about its axis, your weight would be less. If you were in a rotating space habitat that increased its spin rate, you'd "weigh" more. Explain why greater spin rates produce opposite effects in these cases.

You're on the outside of the spinning Earth, but you'd be on the inside of a spinning space habitat. A greater spin rate on the outside of the Earth tends to throw you off a weighing scale, causing it to show a decrease in weight, but against a weighing scale inside the space habitat to show an increase in weight.

centripetal force

a center-directed force that causes an object to follow a curved or circular path F = (mv^2)/r Notice that speed is squared, so twice the speed needs four times the force. The inverse relationship with the radius of curvature tells us that half the radial distance requires twice the force.

torque

a turning or twisting force; the product of force and the lever-arm distance, which tends to produce rotational acceleration torque = lever-arm distance x force

Linear speed is ___ on the outer edge of a rotating object than it is closer to the axis.

greater

Rotational inertia depends on whether the mass is farther or closer to the point of rotation. The farther the mass is, the ____ the rotational inertia.

higher

equilibrium

in general, a state of balance. For mechanical equilibrium, the state in which no net forces and no net torques act. The state in which no net change of energy occurs

tangential speed

linear speed along a curved path; the linear speed of something moving along a circular path is called as such because the direction of motion is tangent to the circumference of the circle

When torques balance each other,

no rotation is produced.

The further weight is from axis,

the harder it is to get moving (higher rotational inertia)

mechanical equilibrium regarding torque

the net torque on a body or system must also be zero for mechanical equilibrium

rotational speed (angular speed)

the number of rotations or revolutions per unit of time; often measured in rotations or revolutions per second or minute

center of mass

the point at the center of an object's mass distribution where all its mass can be considered to be concentrated. For everyday conditions, it is the same as the center of gravity.

center of gravity (CG)

the point at the center of an object's weight distribution, where the force of gravity can be considered to act

angular momentum

the product of a body's rotational inertia and rotational velocity about a particular axis. For an object that is small compared with the radial distance, its magnitude is the product of mass, speed, radial distance from the spin axis. angular momentum = mvr

linear momentum

the product of the mass and velocity of an object; also called momentum

Linear speed and tangenital speed are interchangeable; the unit of these are usually m/s or km/hr

the unit of these are usually m/s or km/hr

Tangential speed ~ radial distance x rotational speed

v ~ rw (v proportional to r omega)