Chapter 5 - Angular Kinematics

Question 1

What is a relative angle?

Answer 1

Relative angle is the angle at a joint formed between the longitudinal axes of adjacent body segments also known as joint angle.
The straight, fully extended position at a joint is regarded as zero degrees

Question 2

What is the straight, fully extended position at a joint regarded as?

Answer 2

Zero degrees

Question 3

Angle at a joint formed between the longitudinal axes of adjacent body segments also known as joint angle the straight, fully extended position at a joint is regarded as zero degrees. True or False?

Answer 3

True.

Question 4

What is an absolute angle?

Answer 4

Absolute angle is the angular orientation of a body segment with respect to a fixed line of reference. The reference lines are usually vertical or horizontal.

Question 5

What is angular displacement?

Answer 5

Angular displacement is the change in angular position.

It is also known as the directed angular distance from initial to final angular position

The vector equivalent of angular distance.

Question 6

What are the units for angular displacement?

Answer 6

It is measured in units of degrees, radians or rotations

Question 7

What is a radian?

Answer 7

The size of the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Question 8

How many degrees and radians is 1/4 revolution?

Answer 8

90 degrees
π/2 radians

Question 9

How many degrees and radians is 1/2 revolution?

Answer 9

180 degrees
π radians

Question 10

How many degrees and radians is 3/4 revolution?

Answer 10

270 degrees
3π/2 radians

Question 11

How many degrees and radians is 1 revolution?

Answer 11

360 degrees
2π radians

Question 12

What is angular velocity?

Answer 12

It is the rate of change in angular position

Question 13

Formula for angular velocity?

Answer 13

angular velocity = angular displacement/time

ω = θ / t

Question 14

What are the units for angular velocity?

Answer 14

degrees/s or radians/s

Question 15

What is angular acceleration?

Answer 15

Angular acceleration is the rate of change in angular velocity

Question 16

What is the formula for angular acceleration?

Answer 16

angular acceleration = Δ angular velocity/time

α = ω2 - ω1 / t

Question 17

What is the unit for angular acceleration?

Answer 17

Angular acceleration is measured in units of deg/s or rad/s

Question 18

Question example:
A golf club is swung with an average angular acceleration of 1.5rad/s^2. What is the angular velocity of the club when it strikes the ball at the end of a 0.8-s swing? (Provide an answer in both radian and degree-based units.)

Answer 18

Acceleration = ω2 - ω1 / t

1.5rad/s^2 = ω2 - ω1 / 0.8

(1.5rad/s^2)(0.8s) = ω2 - 0

ω2 = 1.2 rad/s

In degree-based units:
ω2 = (1.2rad/s)(57.3deg/rad)

ω2 = 68.8 deg/s

Question 19

What is the relationship between linear and angular displacement?

Answer 19

The greater the radius between a given point on a rotating body and the axis of rotation, the greater the linear distance traveled by that point during an angular motion.

Question 20

What is the relationship between linear and angular motion?

Answer 20

The larger the radius of rotation (r), the greater the linear distance (s) traveled by a point on a rotating body.
s = rθ
where θ is the angular displacement in radians.

Question 21

What is the relationship between linear and angular velocity?

Answer 21

Since velocity is displacement over time, linear and angular velocity are related by the same factor that relates displacement: the radius of rotation (r).
v = rω

Question 22

Question example:
Two baseballs are consecutively hit by a bat. The first ball is hit 20cm from the bat’s axis of rotation, and the second bat is hit 40cm from the bat’s axis of rotation. If the angular velocity of the bat was 30rad/s at the instant that both balls were contacted, what was the linear velocity of the bat at the two contact points?

Answer 22

v = rω
For ball 1:

v1 = (0.20m)(30rad/s)
v1 = 6m/s

For ball 2:

v2 = (0.40m)(30rad/s)
v2 = 12m/s

Question 23

What is the relationship between linear and angular acceleration?

Answer 23

The acceleration of a body in angular motion can be resolved into two perpendicular linear acceleration components

Question 24

What is tangential acceleration?

Answer 24

Tangential acceleration is the component of acceleration of angular motion directed along a tangent to the path of motion.
It represents the change in linear speed.
at = v2 - v1 / t

Question 25

What is radial acceleration?

Answer 25

Radial acceleration is the component of acceleration of angular motion directed toward the center of curvature.
It represents change in direction.
ar = v^2/r

Question 26

Question Example:
A windmill-style softball pitcher executes a pitch in 0.65s. If her pitching arm is 0.7m long, what are the magnitudes of the tangential and radial accelerations on the ball just before ball release, when tangential ball speed is 20m/s? What is the magnitude of the total acceleration on the ball at this point?
t = 0.65s
r = 0.7m
v2 = 20m/s

Answer 26

To get tangential acceleration:

at = v2 - v1 / t

Assuming v1 = 0,

at = 20m/s - 0 / 0.65s
at = 30.8m/s^2

To get radial acceleration:
ar = v^2/r

ar = (20m/s)^2 / 0.7m
ar = 571.4m/s

To get total acceleration, perform vector composition of tangential and radial acceleration. Because tangential and radial acceleration are oriented perpendicular to each other, the Pythagorean theorem can be used to calculate the magnitude of total acceleration.

Total acceleration =
√(30.8m/s^2)^2 + (571.4m/s^2)^2 = 572.2m/s^2

Hence, a = 572.2m/s^2