# 6.1 Rotation angle and angular velocity  (Page 3/9)

 Page 3 / 9

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.

## Section summary

• Uniform circular motion is motion in a circle at constant speed. The rotation angle $\text{Δ}\theta$ is defined as the ratio of the arc length to the radius of curvature:
$\text{Δ}\theta =\frac{\text{Δ}s}{r}\text{,}$

where arc length $\text{Δ}s$ is distance traveled along a circular path and $r$ is the radius of curvature of the circular path. The quantity $\text{Δ}\theta$ is measured in units of radians (rad), for which

• The conversion between radians and degrees is $1\phantom{\rule{0.25em}{0ex}}\text{rad}=\text{57}\text{.}3\text{º}$ .
• Angular velocity $\omega$ is the rate of change of an angle,
$\omega =\frac{\text{Δ}\theta }{\text{Δ}t}\text{,}$

where a rotation $\text{Δ}\theta$ takes place in a time $\text{Δ}t$ . The units of angular velocity are radians per second (rad/s). Linear velocity $v$ and angular velocity $\omega$ are related by

## Conceptual questions

There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?

## Problem exercises

Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions—it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?

723 km

Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?

$5×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{rotations}$

(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of $6\text{.}4×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{m}$ at its equator, what is the linear velocity at Earth’s surface?

A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher’s hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?

In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?

A truck with 0.420-m-radius tires travels at 32.0 m/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min?

728 rpm

Integrated Concepts When kicking a football, the kicker rotates his leg about the hip joint.

(a) If the velocity of the tip of the kicker’s shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip’s angular velocity?

(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s?

(c) Find the maximum range of the football, neglecting air resistance.

(b) 500 N

(c) 40.8 m

Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders’ clothing and the wall.

Why is the sky blue...?
It's filtered light from the 2 forms of radiation emitted from the sun. It's mainly filtered UV rays. There's a theory titled Scatter Theory that covers this topic
Mike
A heating coil of resistance 30π is connected to a 240v supply for 5min to boil a quantity of water in a vessel of heat capacity 200jk. If the initial temperature of water is 20°c and it specific heat capacity is 4200jkgk calculate the mass of water in a vessel
A thin equi convex lens is placed on a horizontal plane mirror and a pin held 20 cm vertically above the lens concise in position with its own image the space between the undersurface of d lens and the mirror is filled with water (refractive index =1•33)and then to concise with d image d pin has to
Be raised until its distance from d lens is 27cm find d radius of curvature
Azummiri
what happens when a nuclear bomb and atom bomb bomb explode add the same time near each other
A monkey throws a coconut straight upwards from a coconut tree with a velocity of 10 ms-1. The coconut tree is 30 m high. Calculate the maximum height of the coconut from the top of the coconut tree? Can someone answer my question
v2 =u2 - 2gh 02 =10x10 - 2x9.8xh h = 100 ÷ 19.6 answer = 30 - h.
Ramonyai
why is the north side is always referring to n side of magnetic
who is a nurse
A nurse is a person who takes care of the sick
Bukola
a nurse is also like an assistant to the doctor
explain me wheatstone bridge
good app
samuel
Wheatstone bridge is an instrument used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component.
MUHD
Rockwell Software is Rockwell Automation’s "Retro Encabulator". Now, basically the only new principle involved is that instead of power being generated by the relative motion of conductors and fluxes, it’s produced by the modial interaction of magneto-reluctance and capacitive diractance. The origin
Chip
what refractive index
write a comprehensive note on primary colours
relationship between refractive index, angle of minimum deviation and angle of prism
Harrison
Who knows the formula for binding energy,and what each variable or notation stands for?
1. A black thermocouple measures the temperature in the chamber with black walls.if the air around the thermocouple is 200 C,the walls are at 1000 C,and the heat transfer constant is 15.compute the temperature gradient
what is the relationship between G and g
G is the u. constant, as g stands for grav, accelerate at a discreet point
Mark
Olaiya
pls explain in details
Olaiya
G is a universal constant
Mark
g stands for the gravitational acceleration point. hope this helps you.
Mark
balloon TD is at a gravitational acceleration at a specific point
Mark
I'm sorry this doesn't take dictation very well.
Mark
Can anyone explain the Hooke's law of elasticity?
extension of a spring is proportional to the force applied so long as the force applied does not exceed the springs capacity according to my textbook
Amber
does this help?
Amber
Yes, thanks
Olaiya
so any solid can be compressed how compressed is dependent upon how much force is applied F=deltaL
Amber
sorry, the equation is F=KdeltaL delta is the triangle symbol and L is length so the change in length is proportional to amount of Force applied I believe that is what Hookes law means. anyone catch any mistakes here please correct me :)
Amber
I think it is used only for solids and not liquids, isn't it?
Olaiya
basically as long as you dont exceed the elastic limit the object should return to it original form but if you exceed this limit the object will not return to original shape as it will break
Amber
Thanks for the explanation
Olaiya
yh, liquids don't apply here, that should be viscosity
Chiamaka
hope it helps 😅
Amber
also, an object doesnt have to break necessarily, but it will have a new form :)
Amber
Yes
Olaiya
yeah, I think it is for solids but maybe there is a variation for liquids? that I am not sure of
Amber
ok
Olaiya
good luck!
Amber
Same
Olaiya
aplease i need a help on spcific latent heat of vibrations
Bilgate
specific latent heat of vaporisation
Bilgate
how many kilometers makes a mile
Faizyab
Aakash
equal to 1.609344 kilometers.
MUHD