federica bianco
astro | data science | data for good
PHYS 207.013
Chapter 10
rotation
Instructor: Dr. Bianco
TAs: Joey Betz; Lily Padlow
University of Delaware - Spring 2021
Equity in physics classes
You have the right to the space that you need to succeed based on your identity
Equity in physics classes
How is it possible that the canon of thought in all the disciplines of the Social Sciences and Humanities in the Westernized university [...] is based on the knowledge produced by a few men from five countries in Western Europe (Italy, France, England, Germany and the USA)? How is it possible that men from these five countires achieved such an epistemic privilege to the point that their knowledge today is considered superior over the knowledge of the rest of the world? How did they come to monopolize the authority of knowledge in the world? Why is it that what we know today as social, historical, philosophical, or Critical Theory is based on the socio-historical experience and world views of men from these five countries?
Equity in physics classes
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
ZOOM breakout rooms of 4-5 students
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
the invention of 0 is indian - also mayan arab and chinese - decidedly NOT European
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
In fact while Arabs codified and diffused algebra the Europe was in the "Dark Ages"
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
Nilakantha Somayaji 14 June 1444 – 1544
discovers an infinite series for π.
conceptualizes heliocentric system
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
12th century AD: Jewish polymath Baruch ben Malka, later Abu'l-Barakāt al-Baghdādī, in Iraq formulates a qualitative form of Newton's second law for constant forces.
An explanation of the acceleration of falling bodies as the accumulation of successive increments of power (ref) === Newton laws 300 years early!
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
Until recently, people believed that vulcanization–combining rubber with other materials to make it more durable–was discovered by the American Charles Goodyear in the 19th century. However, historians now think that the Maya were producing rubber products about 3,000 years before Goodyear received his patent in 1843. https://www.history.com/topics/ancient-americas/mayan-scientific-achievements
Decolonizing science
Think about examples of knowledge related to physics that do not come from European-Western culture
HOMEWORK
Equity in physics classes
You have the right to the space that you need to succeed based on your identity
Equity in physics classes
You have the right to the space that you need to succeed based on your identity
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
radians
2 \pi = 360^o
degree
=> radians are revolutions
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
2 \pi = 360^o
angular displacement:
\theta_1
\theta_2
\theta_2 -\theta_1
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
2 \pi = 360^o
\theta_1
\theta_2
angular displacement:
\theta_2 -\theta_1
angular velocity:
\bar{\omega} = \frac{\theta_2 -\theta_1}{t_2 - t_1} = \frac{\Delta \theta}{\Delta t}
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
2 \pi = 360^o
\theta_1
\theta_2
\omega = lim_{\Delta t -> 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}
angular displacement:
\theta_2 -\theta_1
angular velocity:
\bar{\omega} = \frac{\theta_2 -\theta_1}{t_2-t_1} = \frac{\Delta \theta}{\Delta t}
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
2 \pi = 360^o
\theta_1
\theta_2
\alpha = lim_{\Delta t -> 0} \frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt}
angular velocity:
\omega
angular acceleration:
\bar{\alpha} = \frac{\omega -\omega}{t_2 - t_1} = \frac{\Delta \omega}{\Delta t}
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
radians
2 \pi = 360^o
degree
circumference:
2 \pi r
rotation
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
radians
2 \pi = 360^o
degree
circumference:
2 \pi r
linear speed:
v_l = \frac{2 \pi r}{\Delta t}
MATH REVIEW:
trigonometry
vector component: the eq.s of motion you studied in chapter 2 have a parallel here
kinematic of circular motion
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
s = \theta r
\frac{ds}{dt} = \frac{\theta r}{dt} = \frac{d \theta}{dt} r
v~~~~~~=
\omega r
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
s = \theta r
\frac{ds}{dt} = \frac{\theta r}{dt} = \frac{d \theta}{dt} r
v~~~~~~=
\omega r
T = \frac{2\pi r}{v} = \frac{2 \pi}{\omega}
kinematic of circular motion
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
s = \theta r
\frac{ds}{dt} = \frac{\theta r}{dt} = \frac{d \theta}{dt} r
v~~~~~~=
\omega r
T = \frac{2\pi r}{v} = \frac{2 \pi}{\omega}
\frac{ds}{dt} = \frac{d \omega r}{dt} = \frac{d \omega}{dt} r
kinematic of circular motion
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
s = \theta r
\frac{ds}{dt} = \frac{\theta r}{dt} = \frac{d \theta}{dt} r
v~~~~~~=
\omega r
T = \frac{2\pi r}{v} = \frac{2 \pi}{\omega}
\frac{dv}{dt} = \frac{d \omega r}{dt} = \frac{d \omega}{dt} r
a~~~~~~=
\alpha r
kinematic of circular motion
H&R CH10 rotation
\mathrm{angular~ position} ~~\theta = \frac{s}{r} \frac{[L]}{[L]}
radians
s = \theta r
\frac{ds}{dt} = \frac{\theta r}{dt} = \frac{d \theta}{dt} r
v~~~~~~=
\omega r
T = \frac{2\pi r}{v} = \frac{2 \pi}{\omega}
\frac{dv}{dt} = \frac{d \omega r}{dt} = \frac{d \omega}{dt} r
a~~~~~~=
\alpha r
a = \frac{v^2}{r} ={\omega^2 r}
kinematic of circular motion
H&R CH10 rotation
kinematic of circular motion
H&R CH10 rotation
kinematic of circular motion
H&R CH10 rotation
kinematic of circular motion
What is the direction of the rotational velocity vector when you ride a bike?
R
L
F
B
H&R CH10 rotation
K =\frac{1}{2} m_1v_1^2 + \frac{1}{2} m_2v_2^2.... + \frac{1}{2} m_Nv_N^2 = \frac{1}{2}\sum_i m_iv_i^2
energy of rotation
H&R CH10 rotation
K =\frac{1}{2} m_1v_1^2 + \frac{1}{2} m_2v_2^2.... + \frac{1}{2} m_Nv_N^2 = \frac{1}{2}\sum_i m_iv_i^2
K = \frac{1}{2} \sum_i m_ir_i^2 \omega_i^2
energy of rotation
H&R CH10 rotation
K =\frac{1}{2} m_1v_1^2 + \frac{1}{2} m_2v_2^2.... + \frac{1}{2} m_Nv_N^2 = \frac{1}{2}\sum_i m_iv_i^2
K = \frac{1}{2} \sum_i m_ir_i^2 \omega_i^2
moment of inertia
I = \sum_i m_ir_i^2 \\
K = \frac{1}{2}I \omega_i^2\\
energy of rotation
H&R CH10 rotation
K =\frac{1}{2} m_1v_1^2 + \frac{1}{2} m_2v_2^2.... + \frac{1}{2} m_Nv_N^2 = \frac{1}{2}\sum_i m_iv_i^2
K = \frac{1}{2} \sum_i m_ir_i^2 \omega_i^2
I = \sum_i m_i r^2~~\mathrm{or}~~ I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
energy of rotation
moment of inertia
system of particles
solid body
H&R CH10 rotation
K = \frac{1}{2} \sum_i m_ir_i^2 \omega_i^2
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
energy of rotation
H&R CH10 rotation
energy of rotation
K =\frac{1}{2} m_1v_1^2 + \frac{1}{2} m_2v_2^2.... + \frac{1}{2} m_Nv_N^2 = \frac{1}{2}\sum_i m_iv_i^2
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
moment of inertia
the CoM has v=0
K = \frac{1}{2} \sum_i m_ir_i^2 \omega_i^2
H&R CH10 rotation
K =\frac{1}{2} m_1v_1^2 + \frac{1}{2} m_2v_2^2.... + \frac{1}{2} m_Nv_N^2 = \frac{1}{2}\sum_i m_iv_i^2
K = \frac{1}{2} \sum_i m_ir_i^2 \omega_i^2
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
moment of inertia
energy of rotation
H&R CH10 rotation
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
energy of rotation
Parallel Axis Theorem
The moment of inertia of a body is equal to me moment of inertia of the body's center of mass rotation measured along an axis parallel to the axis of rotation, plus the product of the body's mass times the square distance between the axis of rotation and the axis of rotation that passes through the CoM.
I = I_\mathrm{CoM} + Mh^2
H&R CH10 rotation
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
energy of rotation
Parallel Axis Theorem
I = I_\mathrm{CoM} + Mh^2
Let h be the distance between the axis of rotation and the axis parallel to the axis of rotation that goes through the CoM
distance: length of the perpendicular line that joins the two axes
h
Then the moment of inertia I is equal to
H&R CH9 CoM - momentum
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
H&R CH9 CoM - momentum
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
= \int [x^2 + a^2 - 2ax + y^2 + b^2 - 2by] dm =
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
H&R CH9 CoM - momentum
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
= \int [x^2 + a^2 - 2ax + y^2 + b^2 - 2by] dm =
= \int (x^2+y^2)~ dm - 2a \int x ~dm - 2b \int y ~dm + (a^2 + b^2) \int dm
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
H&R CH9 CoM - momentum
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
= \int [x^2 + a^2 - 2ax + y^2 + b^2 - 2by] dm =
= \int (x^2+y^2)~ dm - 2a \int x ~dm - 2b \int y ~dm + (a^2 + b^2) \int dm
I_{CoM}
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
H&R CH9 CoM - momentum
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
= \int [x^2 + a^2 - 2ax + y^2 + b^2 - 2by] dm =
= \int (x^2+y^2)~ dm - 2a \int x ~dm - 2b \int y ~dm + (a^2 + b^2) \int dm
I_{CoM}
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
CoM coords = 0
H&R CH9 CoM - momentum
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
= \int [x^2 + a^2 - 2ax + y^2 + b^2 - 2by] dm =
= \int (x^2+y^2)~ dm - 2a \int x ~dm - 2b \int y ~dm + (a^2 + b^2) \int dm
I_{CoM}
M h^2
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
H&R CH9 CoM - momentum
moment of inertia
I = \int r^2 dm \\
K = \frac{1}{2}I \omega_i^2\\
energy of rotation
I = \int r^2 dm = \int [(x - a)^2 + ( y - b)^2] dm
= \int [x^2 + a^2 - 2ax + y^2 + b^2 - 2by] dm =
= \int (x^2+y^2)~ dm - 2a \int x ~dm - 2b \int y ~dm + (a^2 + b^2) \int dm
I_{CoM}
M h^2
=
+
H&R CH9 CoM - momentum
energy of rotation
H&R CH9 CoM - momentum
energy of rotation
CoM_{tr} = (b/2, h/3)
H&R CH9 CoM - momentum
energy of rotation
CoM_{tr} = (b/2, h/3)
H&R CH9 CoM - momentum
energy of rotation
CoM_{tr} = (b/2, h/3)
H&R CH9 CoM - momentum
energy of rotation
r_{CoM}= 1/M\int r dm
r_{CoM}= \rho/M\int_{y_0(x)}^{y_f(x) }\int_{x_0(y)}^{x_f(y)} \sqrt{x^2 + y^2} dx ~dy
CoM_{tr} = (b/2, h/3)
H&R CH9 CoM - momentum
energy of rotation
CoM
h
energy of rotation
CoM
h
H&R CH9 CoM - momentum
energy of rotation
CoM
M = 34 g\\
h = 34 ~cm
I = I_{CoM} + 0.34*0.34^2 [kg ~m^2]= 3.9 x 10^{-2}
h
H&R CH9 CoM - momentum
energy of rotation
I = I_{CoM} + M \left((\frac{3}{4}L \sin{\theta})^2 + (\frac{1}{2}L \cos{\theta})^2\right)
H&R CH9 CoM - momentum
energy of rotation
I = I_{CoM} + M \left((\frac{3}{4}L \sin{\theta})^2 + (\frac{1}{2}L \cos{\theta})^2\right)
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
torque
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
\theta
\vec{F}
\vec{r}
\tau = rF \sin{90^o} = rF
torque
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
\theta
\vec{F}
\vec{r}
\tau = rF \sin{30^o} = \frac{rF}{2}
torque
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
\vec{F}
\vec{r}
\tau = rF \sin{0^o} = 0
torque
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
\theta
\vec{F}
\vec{r}
line of action
moment arm
torque
torque
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
Case Study: Effect of Handrim Diameter on Performance in a Paralympic Wheelchair Athlete -Gabriel Brizuela Costa, et al. 2009
When greater torque is needed, however, [...] a larger diameter could be more effective. Guo et al., (2006) studied the effect of the diameter of the propulsion handrim (0.54 m, 0.43 m, and 0.32 m) on the propulsive moment generated [...] kinetic, potential, and total mechanical energy delivered [...] with the large handrims were significantly greater than with the small ones. Nevertheless the authors conclude that propelling the with the larger handrim size will have a greater metabolic cost.
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
torque and Force
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
F_t = ma_t\\
\tau = F_t~r = ma_t r\\
particle of mass m at radius r
torque and Force
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
F_t = ma_t\\
\tau = F_t~r = ma_t r\\
a_t = \alpha r\\
\tau = F_t~r = ma_t r = \\m (\alpha r) r = m r^2 \alpha
particle of mass m at radius r
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
torque and Force
Torque is a turning or twisting action on a body about a rotation axis due to a force F
- vector quantity
- has to do with the force F
- has to do with the rotation axis located at some r
F_t = ma_t\\
\tau = F_t~r = ma_t r\\
\alpha = a_t r\\
\tau = F_t~r = ma_t r = \\m (\alpha r) r = m r^2 \alpha
I = mr^2\\
\vec{\tau} = I \vec{\alpha}
particle of mass m at radius r
\vec{\tau} = \vec{F}\times \vec{r} = r F \sin{\theta} ~[N\cdot m]
work and rotational K
Remember:
\Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = W
W = \int_{x_i}^{x_f} F dx
P = \frac{dW}{dt} = F\frac{dx}{dt} = Fv
K = \frac{1}{2}mv^2
K = \frac{1}{2}I\omega^2
For rotation
work and rotational K
Remember:
\Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = W
W = \int_{x_i}^{x_f} F dx
P = \frac{dW}{dt} = F\frac{dx}{dt} = Fv
K = \frac{1}{2}mv^2
K = \frac{1}{2}I\omega^2
\Delta K = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2 = W
For rotation
work and rotational K
Remember:
\Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = W
W = \int_{x_i}^{x_f} F dx
P = \frac{dW}{dt} = F\frac{dx}{dt} = Fv
W = \int_{\theta_i}^{\theta_f} \tau d\theta
K = \frac{1}{2}mv^2
K = \frac{1}{2}I\omega^2
\Delta K = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2 = W
For rotation
work and rotational K
Remember:
\Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = W
W = \int_{x_i}^{x_f} F dx
P = \frac{dW}{dt} = F\frac{dx}{dt} = Fv
W = \int_{\theta_i}^{\theta_f} \tau d\theta
P = \frac{dW}{dt} = \tau\frac{d\theta}{dt} = \tau\omega
K = \frac{1}{2}mv^2
K = \frac{1}{2}I\omega^2
\Delta K = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2 = W
For rotation
phys207 rotation
By federica bianco
phys207 rotation
rotation
