federica bianco PRO
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PHYS 207.013
Chapter 11
rolling motion
Instructor: Dr. Bianco
TAs: Joey Betz; Lily Padlow
University of Delaware - Spring 2021
Decolonizing science
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
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
rolling kinematics
rolling kinematics
s = \theta R\\
v = \omega R\\
v_{CoM} = \omega R
rolling kinematics
rotational inertia abour axis of rotation P
K = \frac{1}{2}I_p\omega^2
P
rolling kinematics
rotational inertia abour axis of rotation P
P
K = \frac{1}{2}I_p\omega^2
Parallel axis theorem
CoM
I_P = I_{CoM} + MR^2
rolling kinematics
rotational inertia abour axis of rotation P
P
K = \frac{1}{2}I_p\omega^2
Parallel axis theorem
CoM
I_P = I_{CoM} + MR^2
K = \frac{1}{2}(I_{CoM} + MR^2)\omega^2 = \\
\frac{1}{2}I_{CoM}\omega^2 + \frac{1}{2}MR^2\omega^2
rolling kinematics
rotational inertia abour axis of rotation P
P
K = \frac{1}{2}I_p\omega^2
Parallel axis theorem
CoM
I_P = I_{CoM} + MR^2
K = \frac{1}{2}(I_{CoM} + MR^2)\omega^2 = \\
\frac{1}{2}I_{CoM}\omega^2 + \frac{1}{2}MR^2\omega^2
v_{CoM} = \omega R
rolling kinematics
rotational inertia abour axis of rotation P
P
K = \frac{1}{2}I_p\omega^2
Parallel axis theorem
CoM
I_P = I_{CoM} + MR^2
K =
\frac{1}{2}I_{CoM}\omega^2 + \frac{1}{2}Mv_{CoM}^2
rolling kinematics
rotational inertia abour axis of rotation P
P
K = \frac{1}{2}I_p\omega^2
Parallel axis theorem
CoM
I_P = I_{CoM} + MR^2
K =
\frac{1}{2}I_{CoM}\omega^2 + \frac{1}{2}Mv_{CoM}^2
K associated to rotation
rolling kinematics
rotational inertia abour axis of rotation P
P
K = \frac{1}{2}I_p\omega^2
Parallel axis theorem
CoM
I_P = I_{CoM} + MR^2
K =
\frac{1}{2}I_{CoM}\omega^2 + \frac{1}{2}Mv_{CoM}^2
K associated to rotation
K associated to translation
rolling kinematics
rolling kinematics: down a ramp
N
mg
fs
maCOM
a_{CoM} = - \frac{g \sin{\theta}}{1+\frac{I_{CoM}}{MR^2}}
rolling kinematics: down a ramp
N
mg
fs
maCOM
\theta = 90^o
a_{CoM} = - \frac{g}{1+\frac{I_{CoM}}{MR_0^2}}
a_{CoM} = - \frac{g \sin{\theta}}{1+\frac{I_{CoM}}{MR^2}}
=>
rolling kinematics: a yoyo
N
mg
fs
maCOM
\theta = 90^o
a_{CoM} = - \frac{g}{1+\frac{I_{CoM}}{MR_0^2}}
a_{CoM} = - \frac{g \sin{\theta}}{1+\frac{I_{CoM}}{MR^2}}
=>
rolling kinematics: a yoyo
rolling kinematics: energy
h
U =
mgh
K =
\frac{1}{2}I_{CoM}\omega^2 + \frac{1}{2}m v_{CoM}^2
v_{h=0, CoM} =
U = \frac{1}{2}\frac{I_{CoM}}{R^2}v_{0,CoM}^2 + \frac{1}{2}m v_{0,CoM}^2
\sqrt{\frac{2mgh}{\frac{I_{CoM}}{R ^2} + m} }
U = \frac{1}{2}v_{0,CoM}^2(\frac{I_{CoM}}{R^2} + m)
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\vec{v}
0
\vec{r}
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\vec{v}
0
\vec{r}=0
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
Angular momentum conservation
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
Angular momentum conservation
\vec{L} = \sum{m_i\vec{r_i}\times\vec{v_i}} = \sum{\vec{l_i}}
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\vec{L} = \sum{\vec{l_i}}
Angular momentum conservation
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
Angular momentum conservation
L_{CoM}
CoM
= const
M
\vec{L} = \sum{\vec{l_i}} = \sum m_i r_i v_i = \sum m_i r_i^2 \omega_i
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\vec{L} = \sum{\vec{l_i}}
Angular momentum is conserved!
L_{CoM}
if the object rotates along an axis that goes through its CoM the Angular Momentum is Conserved (sometimes we call it the "Spin")
= const
Angular momentum
Angular momentum of a rigid body
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\frac{d\vec{v}}{dt} + \frac{d\vec{r}}{dt}\times\vec{v}\right)
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\vec{a} + \vec{v}\times\vec{v}\right)
\frac{d\vec{l}}{dt} =\vec{r}\times m\vec{a} = \tau
\frac{d\vec{L}}{dt} =\sum\vec{r}\times \vec{F} = \tau_{net}
Angular momentum
Angular momentum of a rigid body
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\frac{d\vec{v}}{dt} + \frac{d\vec{r}}{dt}\times\vec{v}\right)
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\vec{a} + \vec{v}\times\vec{v}\right)
\frac{d\vec{l}}{dt} =\vec{r}\times m\vec{a} = \tau
\frac{d\vec{L}}{dt} =\sum\vec{r}\times \vec{F} = \tau_{net}
\theta
\vec{F}
\vec{r}
line of action
moment arm
Angular momentum
Angular momentum of a rigid body
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\frac{d\vec{v}}{dt} + \frac{d\vec{r}}{dt}\times\vec{v}\right)
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\vec{a} + \vec{v}\times\vec{v}\right)
\frac{d\vec{l}}{dt} =\vec{r}\times m\vec{a} = \tau
\frac{d\vec{L}}{dt} =\sum\vec{r}\times \vec{F} = \tau_{net}
\theta
\vec{F}
\vec{r}
moment arm
\vec{F}
Fnet = 0 because of symmetry
line of action
CoM
Angular momentum
Angular momentum of a rigid body
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\frac{d\vec{v}}{dt} + \frac{d\vec{r}}{dt}\times\vec{v}\right)
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\vec{a} + \vec{v}\times\vec{v}\right)
\frac{d\vec{l}}{dt} =\vec{r}\times m\vec{a} = \tau
\frac{d\vec{L}}{dt} =\sum\vec{r}\times \vec{F} = \tau_{net}
Li =\sum \Delta m_i {r_i}\sin{\theta} {v_i} =\\
\sum \Delta m_i ~{r_{i,p} v_i} =
\sum \Delta m_i (\omega r_{i,p} )r_{i,p} =
\omega \Delta m_ir_{i,p}^2 = I\omega
Angular momentum
\vec{l} = \vec{r}\times \vec{p} =
m(\vec{r} \times \vec{v})
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\frac{d\vec{v}}{dt} + \frac{d\vec{r}}{dt}\times\vec{v}\right)
\frac{d\vec{l}}{dt} = m\left(\vec{r}\times\vec{a} + \vec{v}\times\vec{v}\right)
\frac{d\vec{l}}{dt} =\vec{r}\times m\vec{a} = \tau
\frac{d\vec{l}}{dt} =\sum\vec{r}\times \vec{F} = \tau_{net}
\vec{L} = \sum{\vec{l_i}}
\vec{L} = \vec{I}\times\vec{\omega}
The Angular momentum of a system is conserved IF THERE ARE NO EXTERNAL FORCES
Angular momentum
is the reason why we exist!
Angular momentum
is the reason why we exist!
Gyroscope
is it magic??? no: its torque + angular momentum.
Gyroscope
is it magic??? no: its torque + angular momentum.
\vec{\tau} = \frac {d\vec{L}}{dt} \\
\tau = Mgr
\vec{L} = I \vec{\omega} \\
dL = \tau dt \\
d\phi = \frac{dL}{L} = \frac{Mgr dt}{I\omega}
this change in the direction of orientation of the axis of rotation of a body is called
PRECESSION
\\
\Omega = \frac{Mgr}{I\omega}
Gyroscope
is it magic??? no: its torque + angular momentum.
Gyroscope
is it magic??? no: its torque + angular momentum.
Gyroscope
is it magic??? no: its torque + angular momentum.
so the blades can change pitch to provide more lift.
on which blade would you provide the lift to tilt the helicopter forward???
we will talk about it in class on Tuesday!
A
B
C
D
Gyroscope
A
D
Gyroscope
Gyroscope
summary
phys207 rotation energy
By federica bianco
phys207 rotation energy
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