PHYS 218 Chapter 10

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Torque

Vectors at play in torque

(See MATH 152 Chapter 11.3→)

Effectiveness of a force to cause a rotational movement around an axis. Denoted by Greek letter Tau (τ). Measured in "Newton Meters" [Nm] NOT Joules

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tau &= \mathbf{r} \times \mathbf{F} \\ |\tau| &= |\mathbf{r}| |\mathbf{F}| \sin{\theta} \end{align}}

Origin will always be at axis

Torque and Angular Acceleration

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \tau = I\alpha} (equivalent to Newton's second law: $\sum F=ma$

Motion of a Rigid Body

Combination of:

• Translation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2}mv^2} (velocity of center of mass)
• Rotation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2}I\omega^2} (about an axis)

Now when solving problems:

• Free body diagrams must have all forces exactly where they are applied (not necessarily at center of mass of object
• Add the sum of all torques in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{z}} to the sum of all forces in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{x}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{y}}
• Relate the accelerations to each other (assume rope/disk does not slip)

Example

Is the moon rotating? YES! (around earth and on its own axis since same side always faces us)

Rolling without Slipping

Rotation and translation are related:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{cm} = \pm R\omega}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = \pm R\alpha}

(Plus/minus depends on problem; use right-hand rule)

Example

Total Kinetic energy (translational and rotational) of a rolling disk/wheel.

From the center of the disk:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} mv^2 + \frac{1}{2} I\omega^2}

From the point of contact with the ground:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} I_p\omega^2 = \frac{1}{2} \left(I_{cm} +mR^2 \right) \omega^2}

These two formulas are the same when simplified.

Problem

Disk rolling down an inclined plane

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \sqrt{\frac{2gh}{1+\frac{I}{mR^2}}} = \sqrt{\frac{2gh}{1+c}}}

Work done by Torque

Work done by a force applied to a rotating object (W is still measured in Joules):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \int_{\theta_1}^{\theta_2} \tau \mathrm{d}\theta}

Corollary

Work of net force:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_N = \frac{1}{2} I \omega_f^2 - \frac{1}{2} I \omega_i^2}

Power of Torque

(Use second equation if torque is constant.)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} P &= \frac{\Delta W}{\Delta t} = \frac{\mathrm{d}}{\mathrm{d}t} \int_{\theta_1}^{\theta_2} \tau \mathrm{d}\theta \\ &= \tau \omega \end{align}}

Angular Momentum

Angular momentum is represented by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_q} represent the position with respect to the coordinate origin point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}} is the cross product of the position vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} 's linear momentum:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{L}_q & = \mathbf{r}_q \times \mathbf{P} = \mathbf{r}_q \times m\mathbf{v} \\ L &= rmv \end{align}}

Note. Position vector can be broken into components parallel and perpendicular to velocity vector. The magnitude of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}} only depends on the perpendicular component

Note

Angular Momentum if spinning around its center of mass for any rigid body is completely independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} !

Object has intrinsic angular momentum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L} = I_{cm} \mathbf{\omega}}

Conservation of Angular Momentum

If there is no external torque applied to a rotating body, then the angular momentum does not change and is therefore conserved.

This also applies to each component: no ext. torque in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{z}} means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} component of angular momentum remains the same.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{i,q} = L_{f,q}\,\!}

• Gravity/weight performs no work and thus angular momentum is conserved.

Example

A small disk attached to a big disk Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} meters away from the big disk's axis, both of which are spinning with a positive angular velocity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = I_s\omega_s + (I_s+M_sd^2) \omega_b}