MHB Disc Rotation: Mass m & Radius r Down Slope of tan^-1 3/4

markosheehan
Messages
133
Reaction score
0
a disc of mass m and radius r rolls down a slope of incline tan^-1 3/4. the slope is rough enough to prevent slipping. the disc travels from rest a distance of s metres straight down the slope.
(i) show that the linear acceleration of the disc is 2/5 g.
what i did was i first revolved the forces and i got 2 equations
R=4/5 g and 3/5 mg-F=ma but i don't know where to from here
 
Mathematics news on Phys.org
Okay, if we recognize that the net force $F=Ma_{\text{CM}}$ on the disk parallel to the incline is the component of the disk's weight parallel to the incline less the force of friction $f$ keeping the disk from slipping, we may write (letting $\theta$ be the angle of the incline above the horizontal):

$$Mg\sin(\theta)-f=Ma_{\text{CM}}$$

The force of friction results in a torque about the center of mass, having a lever arm $R$, so we may write (where $I$ is the moment of inertia for the disk):

$$\tau=Rf=I\alpha$$

Solving for $f$, there results:

$$f=\frac{I\alpha}{R}$$

And so we now have:

$$Mg\sin(\theta)-\frac{I\alpha}{R}=Ma_{\text{CM}}$$

Now, we may relate the angular acceleration $\alpha$ of the disk to its linear acceleration$a$ as follows:

$$a=R\alpha$$

Since the disk rolls without slipping, we know:

$$a=a_{\text{CM}}$$

Hence:

$$\alpha=\frac{a_{\text{CM}}}{R}$$

And now we have:

$$Mg\sin(\theta)-\frac{I\dfrac{a_{\text{CM}}}{R}}{R}=Ma_{\text{CM}}$$

Or:

$$Mg\sin(\theta)-\frac{Ia_{\text{CM}}}{R^2}=Ma_{\text{CM}}$$

So, what do you get when you solve for $a_{\text{CM}}$?
 
i get 2/5 g when i solve for a but i don't get your full solution. what is the formula of a disc for angular acceleration what does a=Ra stand for.
 
For a rolling object (circular such as a cylinder, sphere, disk, hoop, etc.) of radius $R$, the linear acceleration $a$ and the angular acceleration $\alpha$ are related as follows:

$$a=R\alpha$$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top