# 10.5 Angular momentum and its conservation  (Page 2/7)

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## Calculating the torque in a kick

The person whose leg is shown in [link] kicks his leg by exerting a 2000-N force with his upper leg muscle. The effective perpendicular lever arm is 2.20 cm. Given the moment of inertia of the lower leg is $1.25 kg\cdot {\text{m}}^{2}$ , (a) find the angular acceleration of the leg. (b) Neglecting the gravitational force, what is the rotational kinetic energy of the leg after it has rotated through $\text{57}\text{.}3º$ (1.00 rad)?

Strategy

The angular acceleration can be found using the rotational analog to Newton’s second law, or $\alpha =\text{net}\phantom{\rule{0.25em}{0ex}}\tau /I$ . The moment of inertia $I$ is given and the torque can be found easily from the given force and perpendicular lever arm. Once the angular acceleration $\alpha$ is known, the final angular velocity and rotational kinetic energy can be calculated.

Solution to (a)

From the rotational analog to Newton’s second law, the angular acceleration $\alpha$ is

$\alpha =\frac{\text{net}\phantom{\rule{0.25em}{0ex}}\tau }{I}.$

Because the force and the perpendicular lever arm are given and the leg is vertical so that its weight does not create a torque, the net torque is thus

$\begin{array}{lll}\text{net}\phantom{\rule{0.25em}{0ex}}\tau & =& {r}_{\perp }F\\ & =& \left(0\text{.}\text{0220 m}\right)\left(\text{2000}\phantom{\rule{0.25em}{0ex}}\text{N}\right)\\ & =& \text{44}\text{.}\text{0 N}\cdot \text{m.}\end{array}$

Substituting this value for the torque and the given value for the moment of inertia into the expression for $\alpha$ gives

$\alpha =\frac{\text{44}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{N}\cdot \text{m}}{1\text{.}\text{25}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}}=\text{35}\text{.}2\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}.$

Solution to (b)

The final angular velocity can be calculated from the kinematic expression

${\omega }^{2}={{\omega }_{0}}^{2}+2\text{αθ}$

or

${\omega }^{2}=2\text{αθ}$

because the initial angular velocity is zero. The kinetic energy of rotation is

${\text{KE}}_{\text{rot}}=\frac{1}{2}{\mathrm{I\omega }}^{2}$

so it is most convenient to use the value of ${\omega }^{2}$ just found and the given value for the moment of inertia. The kinetic energy is then

$\begin{array}{lll}{\text{KE}}_{\text{rot}}& =& 0.5\left(1\text{.25}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}\right)\left(\text{70.}4\phantom{\rule{0.25em}{0ex}}{\text{rad}}^{2}/{\text{s}}^{2}\right)\\ & =& \text{44}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}.$

Discussion

These values are reasonable for a person kicking his leg starting from the position shown. The weight of the leg can be neglected in part (a) because it exerts no torque when the center of gravity of the lower leg is directly beneath the pivot in the knee. In part (b), the force exerted by the upper leg is so large that its torque is much greater than that created by the weight of the lower leg as it rotates. The rotational kinetic energy given to the lower leg is enough that it could give a ball a significant velocity by transferring some of this energy in a kick.

## Making connections: conservation laws

Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero.

## Conservation of angular momentum

We can now understand why Earth keeps on spinning. As we saw in the previous example, $\text{Δ}L=\left(\text{net}\phantom{\rule{0.25em}{0ex}}\tau \right)\text{Δ}t$ . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.

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it's the time rate of change of distance
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distance in a given direction is diplacement
Musa
Distance in a spacified direction
you shouldn't say distance,displacement and distance are two different things .distance can be lopped curved but displacement is always in a straight line so you can't use distance to define it. displacement is the change of position in a specified direction.
Joshua
Well stayed josh👍
Joshua
well explained
Mary
what is the meaning of physics
to study objects in motion and how they interact or take part in the natural phenomenon of the universe.
Phill
an object that has a small mass and an object has a large mase have the same momentum which has high kinetic energy
The with smaller mass
how
Faith
Since you said they have the same momentum.. So meaning that there is more like an inverse proportionality in the quantities used to find the momentum. We are told that the the is a larger mass and a smaller mass., so we can conclude that the smaller mass had higher velocity as compared to other one
Mathamaticaly correct
Mathmaticaly correct :)
I have proven it by using my own values
Larger mass=4g Smaller mass=2g Momentum of both=8 Meaning V for L =2 and V for S=4 Now find there kinetic energies using the data presented
grateful soul...thanks alot
Faith
Welcome
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30s
how can i calculate it's height
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is speed the same as velocity
no
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in a question i ought to find the momentum but was given just mass and speed
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just multiply mass and speed then you have the magnitude of momentem
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Yes
Consider speed to be velocity
it worked our . . thanks
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I think you shorten am not sure
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shorten it, since that is practice able using the simple pendulum as experiment
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it'll always give the results needed no need to adjust the length, it is always measured by the starting time and ending time by the clock
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it's not in relation to other clocks
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wat is d formular for newton's third principle
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okay
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that resistance is not measured yet, it may be probably in the next generation of scientists
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Is fundamental quantities under physical quantities?
please I didn't not understand the concept of the physical therapy
physiotherapy - it's a practice of exercising for healthy living.
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this is not in this book, it's from other experiences.
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am new in the group
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