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ludmilkaskok [199]
3 years ago
11

Is mercury a terrestrial or gaseous planet

Physics
1 answer:
liraira [26]3 years ago
8 0

Mercury a terrestrial. It isn't made of gas.

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A cylinder of radius R, length L, and mass M is released from rest on a slope inclined at angle θ. It is oriented to roll straig
inna [77]

Answer:

\mu_s=\frac{1}{3}\tan \theta

Explanation:

Let the minimum coefficient of static friction be \mu_s.

Given:

Mass of the cylinder = M

Radius of the cylinder = R

Length of the cylinder = L

Angle of inclination = \theta

Initial velocity of the cylinder (Released from rest) = 0

Since, the cylinder is translating and rolling down the incline, it has both translational and rotational motion. So, we need to consider the effect of moment of Inertia also.

We know that, for a rolling object, torque acting on it is given as the product of moment of inertia and its angular acceleration. So,

\tau =I\alpha

Now, angular acceleration is given as:

\alpha = \frac{a}{R}\\Where, a\rightarrow \textrm{linear acceleration of the cylinder}

Also, moment of inertia for a cylinder is given as:

I=\frac{MR^2}{2}

Therefore, the torque acting on the cylinder can be rewritten as:

\tau = \frac{MR^2}{2}\times \frac{a}{R}=\frac{MRa}{2}------ 1

Consider the free body diagram of the cylinder on the incline. The forces acting along the incline are mg\sin \theta\ and\ f. The net force acting along the incline is given as:

F_{net}=Mg\sin \theta-f\\But,\ f=\mu_s N\\So, F_{net}=Mg\sin \theta -\mu_s N-------- 2

Now, consider the forces acting perpendicular to the incline. As there is no motion in the perpendicular direction, net force is zero.

So, N=Mg\cos \theta

Plugging in N=Mg\cos \theta in equation (2), we get

F_{net}=Mg\sin \theta -\mu_s Mg\cos \theta\\F_{net}=Mg(\sin \theta-\mu_s \cos \theta)--------------3

Now, as per Newton's second law,

F_{net}=Ma\\Mg(\sin \theta-\mu_s \cos \theta)=Ma\\\therefore a=g(\sin \theta-\mu_s \cos \theta)------4

Now, torque acting on the cylinder is provided by the frictional force and is given as the product of frictional force and radius of the cylinder.

\tau=fR\\\frac{MRa}{2}=\mu_sMg\cos \theta\times  R\\\\a=2\times \mu_sg\cos \theta\\\\But, a=g(\sin \theta-\mu_s \cos \theta)\\\\\therefore g(\sin \theta-\mu_s \cos \theta)=2\times \mu_sg\cos \theta\\\\\sin \theta-\mu_s \cos \theta=2\mu_s\cos \theta\\\\\sin \theta=2\mu_s\cos \theta+\mu_s\cos \theta\\\\\sin \theta=3\mu_s \cos \theta\\\\\mu_s=\frac{\sin \theta}{3\cos \theta}\\\\\mu_s=\frac{1}{3}\tan \theta............(\because \frac{\sin \theta}{\cos \theta}=\tan \theta)

Therefore, the minimum coefficient of static friction needed for the cylinder to roll down without slipping is given as:

\mu_s=\frac{1}{3}\tan \theta

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thermal equilibrium is reached when both objects are the same temperature.
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