Unit 8 Linear and Angular Motions

which rotation direction is positive?

Counterclockwise is positive. Clockwise is negative.

How does the body often produce linear motion?

Through rotation of body segments around joints.

What is the radius of rotation?

The distance from the axis of rotation to the point being analyzed.

What is arc length?

The distance traveled along a curved path.

What is the equation for arc length during a one-direction rotation?

s = r|Δθ|, with Δθ in radians.

What is chord length?

The straight-line distance between two points on a circle.

What is the equation for chord length?

d = 2r sin(φ/2), where φ is the smaller central angle between the endpoints.

For a circular arc, which is generally longer: arc length or chord length?

Arc length

A point completes one full revolution. What are its distance traveled and displacement?

Distance is 2πr. Displacement is zero.

Why must degrees be converted before using s = r|Δθ|?

The equation requires the angle in radians.

A point rotates through 90° at a radius of 2 m. What arc length does it travel?

π m, or approximately 3.14 m.

A circle has radius 10 cm and central angle 23°. What is the arc length?

Approximately 4.01 cm.

A runner travels 39.27 m along a circle of radius 25 m. What angle is covered?

Approximately 1.571 radians, or 90°.

An ankle rotates through 1.2 radians at a radius of 45 cm. What arc length does it travel?

54 cm.

At the same 45 cm radius, how much farther does the ankle travel through 1.6 radians than 1.2 radians?

18 cm farther.

Two points rotate through the same angle. Which travels farther?

The point farther from the axis.

Two runners stay side by side around a turn. Who must travel farther?

The runner in the outside lane.

Two runners stay side by side around a turn. Who needs greater linear speed?

The outside runner.

For a semicircular turn, how much extra distance does a runner cover at a radius 1 m larger?

π m, or approximately 3.14 m.

What is tangential velocity?

The linear velocity of a rotating point relative to its axis.

Which direction does tangential velocity point?

Tangent to the circular path, perpendicular to the radius.

What is the relationship between tangential speed and angular speed?

|vT| = r|ω|.

Which angular velocity units are required in vT = rω?

Radians per second.

A forearm rotates at 4 rad/s with the hand 0.35 m from the elbow. What is hand speed?

1.4 m/s.

A foot moves tangentially at 12 m/s at a radius of 0.9 m. What is angular speed?

Approximately 13.3 rad/s.

Two points on a rigid rotating bat share what angular quantity?

Angular velocity.

Do all points on a rigid rotating bat have the same tangential speed?

No. Points farther from the axis move faster.

At the same angular velocity, what happens to tangential speed when radius doubles?

It doubles.

At the same radius, what happens to tangential speed when angular speed doubles?

It doubles.

At the axis of rotation, what is tangential speed relative to the axis?

Zero.

A point is at angular position θ and rotates counterclockwise. What is its tangential direction?

θ + 90°.

A point is at angular position θ and rotates clockwise. What is its tangential direction?

θ − 90°.

A point is directly right of the axis and rotates counterclockwise. Which way is its velocity?

Upward.

A point is directly above the axis and rotates clockwise. Which way is its velocity?

To the right.

A bat rotates at 600°/s. What is its angular speed in rad/s?

Approximately 10.47 rad/s.

At 600°/s, what are tangential speeds at radii of 0.40 m and 0.80 m?

Approximately 4.19 m/s and 8.38 m/s.

Does a longer radius guarantee greater endpoint speed in an actual human movement?

No. The comparison also depends on angular velocity.

What did the grip study report about choking up on the bat?

Shorter swing and stride times, greater bat-tip velocity, and no detected accuracy difference.

Why can choking up produce greater bat-tip speed despite a shorter radius?

A sufficiently greater angular velocity can offset the shorter radius.

What are the two acceleration components in circular motion?

Tangential and radial acceleration.

What does tangential acceleration change?

Tangential speed.

What does radial acceleration change?

The direction of velocity.

Which way does radial acceleration point?

Inward toward the axis of rotation.

Which way does tangential acceleration point when a rotating point speeds up?

Along its tangential velocity.

Which way does tangential acceleration point when a rotating point slows down?

Opposite its tangential velocity.

What is average tangential acceleration for fixed-radius rotation?

aT = ΔvT/Δt, using signed tangential speed.

How are tangential and angular acceleration related at fixed radius?

aT = rα.

Which angular acceleration units are required in aT = rα?

Radians per second squared.

What are the equations for radial acceleration magnitude?

aR = vT²/r = rω².

Can radial acceleration be present when angular acceleration is zero?

Yes. Constant-speed circular motion still changes velocity direction.

At constant speed around a circle, what are the tangential and radial accelerations?

Tangential acceleration is zero. Radial acceleration points inward.

Does clockwise rotation make radial acceleration point outward?

No. It points inward for either rotation direction.

At fixed radius, what happens to radial acceleration when angular speed doubles?

It quadruples.

At fixed angular speed, what happens to radial acceleration when radius doubles?

It doubles.

At fixed tangential speed, what happens to radial acceleration when radius doubles?

It halves.

How do you find total acceleration magnitude from perpendicular tangential and radial components?

|a| = √(aT² + aR²).

Why can’t you simply add tangential and radial acceleration magnitudes?

They point in perpendicular directions.

What direction does a released rotating object initially travel?

Tangent to its path at the release point.

Does the velocity of a hammer ball suddenly change at the instant of release?

No. Its initial release velocity matches its velocity immediately before release.

Does a released hammer ball have zero acceleration?

No. Gravity still accelerates it downward.