From
[link] (a), we see the force vectors involved in preventing the wheel from slipping. In (b), point
P that touches the surface is at rest relative to the surface. Relative to the center of mass, point
P has velocity
$\text{\u2212}R\omega \widehat{i}$ , where
R is the radius of the wheel and
$\omega $ is the wheel’s angular velocity about its axis. Since the wheel is rolling, the velocity of
P with respect to the surface is its velocity with respect to the center of mass plus the velocity of the center of mass with respect to the surface:
Since the velocity of
P relative to the surface is zero,
${v}_{P}=0$ , this says that
${v}_{\text{CM}}=R\omega .$
Thus, the velocity of the wheel’s center of mass is its radius times the angular velocity about its axis. We show the correspondence of the linear variable on the left side of the equation with the angular variable on the right side of the equation. This is done below for the linear acceleration.
If we differentiate
[link] on the left side of the equation, we obtain an expression for the linear acceleration of the center of mass. On the right side of the equation,
R is a constant and since
$\alpha =\frac{d\omega}{dt},$ we have
${a}_{\text{CM}}=R\alpha .$
Furthermore, we can find the distance the wheel travels in terms of angular variables by referring to
[link] . As the wheel rolls from point
A to point
B , its outer surface maps onto the ground by exactly the distance travelled, which is
${d}_{\text{CM}}.$ We see from
[link] that the length of the outer surface that maps onto the ground is the arc length
$R\theta \text{}$ . Equating the two distances, we obtain
${d}_{\text{CM}}=R\theta .$
Rolling down an inclined plane
A solid cylinder rolls down an inclined plane without slipping, starting from rest. It has mass
m and radius
r . (a) What is its acceleration? (b) What condition must the coefficient of static friction
${\mu}_{\text{S}}$ satisfy so the cylinder does not slip?
Strategy
Draw a sketch and free-body diagram, and choose a coordinate system. We put
x in the direction down the plane and
y upward perpendicular to the plane. Identify the forces involved. These are the normal force, the force of gravity, and the force due to friction. Write down Newton’s laws in the
x - and
y -directions, and Newton’s law for rotation, and then solve for the acceleration and force due to friction.
Solution
The free-body diagram and sketch are shown in
[link] , including the normal force, components of the weight, and the static friction force. There is barely enough friction to keep the cylinder rolling without slipping. Since there is no slipping, the magnitude of the friction force is less than or equal to
${\mu}_{S}N$ . Writing down Newton’s laws in the
x - and
y -directions, we have
The torques are calculated about the axis through the center of mass of the cylinder. The only nonzero torque is provided by the friction force. We have
${f}_{\text{S}}r={I}_{\text{CM}}\alpha .$
Finally, the linear acceleration is related to the angular acceleration by
${({a}_{\text{CM}})}_{x}=r\alpha .$
These equations can be used to solve for
${a}_{\text{CM}},\alpha ,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{f}_{\text{S}}$ in terms of the moment of inertia, where we have dropped the
x -subscript. We write
${a}_{\text{CM}}$ in terms of the vertical component of gravity and the friction force, and make the following substitutions.
Note that this result is independent of the coefficient of static friction,
${\mu}_{\text{S}}$ .
Since we have a solid cylinder, from
[link] , we have
${I}_{\text{CM}}=m{r}^{2}\text{/}2$ and
Substituting this expression into the condition for no slipping, and noting that
$N=mg\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta $ , we have
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7 m/s2 .
At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at 30.0° 30.0° with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.
vector has both magnitude & direction but scalar has only magnitude
Manash
my marunong ba dto mag prove ng geometry
ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
SO ASK
What is Newtonian mechanics in physics?
Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
Bhaskar
tq u.
Prema
these are the best physics books one can fond both theory and applications.
Bhaskar
can any one suggest best book for maths with lot of Tricks?
Vivek
what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here...
.
1. Newton's laws of Motion, are they applicable to motions of all speeds?
2.state the speeds which are applicable to Newtons laws of Motion
Derek
mechanics which follows Newtons law
Manash
The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
jyotirmayee
prove Newton's first law of motion
prince
Hello frnds what is physics in general?
Ngeh
A block of mass m is attached to a spring with spring constant k and free to slide along a horizontal frictionless surface. At t=0, the block spring system is stretched on amount x>0 from the equilibrium position and is released from rest Vx = 0
What is the period of oscillation of the block?
What
Ella
What is the velocity of the block when it first comes back to the equilibrium position?