# 11.1 Rolling motion  (Page 5/6)

 Page 5 / 6
${v}_{\text{CM}}=\sqrt{\left(3.71\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}\right)25.0\phantom{\rule{0.2em}{0ex}}\text{m}}=9.63\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}\text{.}$

## Significance

This is a fairly accurate result considering that Mars has very little atmosphere, and the loss of energy due to air resistance would be minimal. The result also assumes that the terrain is smooth, such that the wheel wouldn’t encounter rocks and bumps along the way.

Also, in this example, the kinetic energy, or energy of motion, is equally shared between linear and rotational motion. If we look at the moments of inertia in [link] , we see that the hollow cylinder has the largest moment of inertia for a given radius and mass. If the wheels of the rover were solid and approximated by solid cylinders, for example, there would be more kinetic energy in linear motion than in rotational motion. This would give the wheel a larger linear velocity than the hollow cylinder approximation. Thus, the solid cylinder would reach the bottom of the basin faster than the hollow cylinder.

## Summary

• In rolling motion without slipping, a static friction force is present between the rolling object and the surface. The relations ${v}_{\text{CM}}=R\omega ,{a}_{\text{CM}}=R\alpha ,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{d}_{\text{CM}}=R\theta$ all apply, such that the linear velocity, acceleration, and distance of the center of mass are the angular variables multiplied by the radius of the object.
• In rolling motion with slipping, a kinetic friction force arises between the rolling object and the surface. In this case, ${v}_{\text{CM}}\ne R\omega ,{a}_{\text{CM}}\ne R\alpha ,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{d}_{\text{CM}}\ne R\theta$ .
• Energy conservation can be used to analyze rolling motion. Energy is conserved in rolling motion without slipping. Energy is not conserved in rolling motion with slipping due to the heat generated by kinetic friction.

## Conceptual questions

Can a round object released from rest at the top of a frictionless incline undergo rolling motion?

No, the static friction force is zero.

A cylindrical can of radius R is rolling across a horizontal surface without slipping. (a) After one complete revolution of the can, what is the distance that its center of mass has moved? (b) Would this distance be greater or smaller if slipping occurred?

A wheel is released from the top on an incline. Is the wheel most likely to slip if the incline is steep or gently sloped?

The wheel is more likely to slip on a steep incline since the coefficient of static friction must increase with the angle to keep rolling motion without slipping.

Which rolls down an inclined plane faster, a hollow cylinder or a solid sphere? Both have the same mass and radius.

A hollow sphere and a hollow cylinder of the same radius and mass roll up an incline without slipping and have the same initial center of mass velocity. Which object reaches a greater height before stopping?

The cylinder reaches a greater height. By [link] , its acceleration in the direction down the incline would be less.

## Problems

What is the angular velocity of a 75.0-cm-diameter tire on an automobile traveling at 90.0 km/h?

${v}_{\text{CM}}=R\omega \phantom{\rule{0.2em}{0ex}}⇒\omega =66.7\phantom{\rule{0.2em}{0ex}}\text{rad/s}$

A boy rides his bicycle 2.00 km. The wheels have radius 30.0 cm. What is the total angle the tires rotate through during his trip?

If the boy on the bicycle in the preceding problem accelerates from rest to a speed of 10.0 m/s in 10.0 s, what is the angular acceleration of the tires?

$\alpha =3.3\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}{\text{s}}^{2}$

Formula One race cars have 66-cm-diameter tires. If a Formula One averages a speed of 300 km/h during a race, what is the angular displacement in revolutions of the wheels if the race car maintains this speed for 1.5 hours?

A marble rolls down an incline at $30\text{°}$ from rest. (a) What is its acceleration? (b) How far does it go in 3.0 s?

${I}_{\text{CM}}=\frac{2}{5}m{r}^{2},\phantom{\rule{0.2em}{0ex}}{a}_{\text{CM}}=3.5\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2};\phantom{\rule{0.2em}{0ex}}x=15.75\phantom{\rule{0.2em}{0ex}}\text{m}$

Repeat the preceding problem replacing the marble with a solid cylinder. Explain the new result.

A rigid body with a cylindrical cross-section is released from the top of a $30\text{°}$ incline. It rolls 10.0 m to the bottom in 2.60 s. Find the moment of inertia of the body in terms of its mass m and radius r.

positive is down the incline plane;
${a}_{\text{CM}}=\frac{mg\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta }{m+\left({I}_{\text{CM}}\text{/}{r}^{2}\right)}⇒{I}_{\text{CM}}={r}^{2}\left[\frac{mg\phantom{\rule{0.2em}{0ex}}\text{sin}30}{{a}_{\text{CM}}}-m\right]$ ,
$x-{x}_{0}={v}_{0}t-\frac{1}{2}{a}_{\text{CM}}{t}^{2}⇒{a}_{\text{CM}}=2.96\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2},$
${I}_{\text{CM}}=0.66\phantom{\rule{0.2em}{0ex}}m{r}^{2}$

A yo-yo can be thought of a solid cylinder of mass m and radius r that has a light string wrapped around its circumference (see below). One end of the string is held fixed in space. If the cylinder falls as the string unwinds without slipping, what is the acceleration of the cylinder?

A solid cylinder of radius 10.0 cm rolls down an incline with slipping. The angle of the incline is $30\text{°}.$ The coefficient of kinetic friction on the surface is 0.400. What is the angular acceleration of the solid cylinder? What is the linear acceleration?

$\alpha =67.9\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}{\text{s}}^{2}$ ,
${\left({a}_{\text{CM}}\right)}_{x}=1.5\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}$

A bowling ball rolls up a ramp 0.5 m high without slipping to storage. It has an initial velocity of its center of mass of 3.0 m/s. (a) What is its velocity at the top of the ramp? (b) If the ramp is 1 m high does it make it to the top?

A 40.0-kg solid cylinder is rolling across a horizontal surface at a speed of 6.0 m/s. How much work is required to stop it?

$W=-1080.0\phantom{\rule{0.2em}{0ex}}\text{J}$

A 40.0-kg solid sphere is rolling across a horizontal surface with a speed of 6.0 m/s. How much work is required to stop it? Compare results with the preceding problem.

A solid cylinder rolls up an incline at an angle of $20\text{°}.$ If it starts at the bottom with a speed of 10 m/s, how far up the incline does it travel?

Mechanical energy at the bottom equals mechanical energy at the top;
$\frac{1}{2}m{v}_{0}^{2}+\frac{1}{2}\left(\frac{1}{2}m{r}^{2}\right){\left(\frac{{v}_{0}}{r}\right)}^{2}=mgh⇒h=\frac{1}{g}\left(\frac{1}{2}+\frac{1}{4}\right){v}_{0}^{2}$ ,
$h=7.7\phantom{\rule{0.2em}{0ex}}\text{m,}$ so the distance up the incline is $22.5\phantom{\rule{0.2em}{0ex}}\text{m}$ .

A solid cylindrical wheel of mass M and radius R is pulled by a force $\stackrel{\to }{F}$ applied to the center of the wheel at $37\text{°}$ to the horizontal (see the following figure). If the wheel is to roll without slipping, what is the maximum value of $|\stackrel{\to }{F}|?$ The coefficients of static and kinetic friction are ${\mu }_{\text{S}}=0.40\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\mu }_{\text{k}}=0.30.$

A hollow cylinder is given a velocity of 5.0 m/s and rolls up an incline to a height of 1.0 m. If a hollow sphere of the same mass and radius is given the same initial velocity, how high does it roll up the incline?

Use energy conservation
$\frac{1}{2}m{v}_{0}^{2}+\frac{1}{2}{I}_{\text{Cyl}}{\omega }_{0}^{2}=mg{h}_{\text{Cyl}}$ ,
$\frac{1}{2}m{v}_{0}^{2}+\frac{1}{2}{I}_{\text{Sph}}{\omega }_{0}^{2}=mg{h}_{\text{Sph}}$ .
Subtracting the two equations, eliminating the initial translational energy, we have
$\frac{1}{2}{I}_{\text{Cyl}}{\omega }_{0}^{2}-\frac{1}{2}{I}_{\text{Sph}}{\omega }_{0}^{2}=mg\left({h}_{\text{Cyl}}-{h}_{\text{Sph}}\right)$ ,
$\frac{1}{2}m{r}^{2}{\left(\frac{{v}_{0}}{r}\right)}^{2}-\frac{1}{2}\frac{2}{3}m{r}^{2}{\left(\frac{{v}_{0}}{r}\right)}^{2}=mg\left({h}_{\text{Cyl}}-{h}_{\text{Sph}}\right)$ ,
$\frac{1}{2}{v}_{0}^{2}-\frac{1}{2}\frac{2}{3}{v}_{0}^{2}=g\left({h}_{\text{Cyl}}-{h}_{\text{Sph}}\right)$ ,
${h}_{\text{Cyl}}-{h}_{\text{Sph}}=\frac{1}{g}\left(\frac{1}{2}-\frac{1}{3}\right){v}_{0}^{2}=\frac{1}{9.8\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}}\left(\frac{1}{6}\right)\left(5.0\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s)}}^{2}=0.43\phantom{\rule{0.2em}{0ex}}\text{m}$ .
Thus, the hollow sphere, with the smaller moment of inertia, rolls up to a lower height of $1.0-0.43=0.57\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$

what is electromagnetism
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
units as in how
praise
What is th formular for force
F = m x a
Santos
State newton's second law of motion
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
how can one calculate the value of a given quantity
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
What is physics
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
what is thought to be the energy density of multiverse and is the space between universes really space
tibebeab
can you explain it
Guhan
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
on the mid point
Suzana
is the magnetic field of the earth changing?
tibebeab
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab
A stone is dropped into a well of 19.6m deep and the impact of sound heared after 2.056 second ,find the velocity of sound in air.
9.53 m/s ?
Kyla
In this case, the velocity of sound is 350 m/s.
Zahangir
why?
Kyla
some calculations is need. then you will get exact result.
Zahangir
i mean how? isn't it just a d over t?
Kyla
calculate the time it takes the stone to hit the ground then minus the stone's time to the total time... then divide the total distance by the difference of the time
Snuggly
awit lenard. Hahahah ari ga to!
Kyla