Why Does a Sphere Start Skidding on a Dome at a 30 Degree Angle?

[Wen]

Qn
A uniform Solid Sphere, placed on top of a spherical dome, rolls down the dome with negligible initial velocity. The sphere sphere start to skid(sliding at the point of contact )when the angle( angle btw the line joining the centre of the dome to the centre of the sphere and the verticle) equal to 30 deg.
Calculate the coefficient of static friction between the sphere and the dome.
I really do not know how to start. Can comeone please help me?

 

[quantumdude]

Start by drawing a free body diagram of the sphere on the dome, and applying Newton's second law.

What forces are acting? What is the expression for the acceleration of the rolling sphere?

 

[quicknote]

I would first like to clarify the concept of rotational dynamics. Rotational dynamics is the study of the motion of rotating objects and the forces that act upon them. It involves understanding the relationship between rotational motion, angular velocity, and angular acceleration.

In this scenario, we have a uniform solid sphere placed on top of a spherical dome, which is essentially a curved surface. When the sphere is released from rest, it starts to roll down the dome. This is due to the force of gravity acting on the sphere, causing it to accelerate down the dome.

However, at a certain point, when the angle between the line joining the centre of the dome and the centre of the sphere and the vertical is 30 degrees, the sphere starts to skid. This means that the sphere is no longer rolling but is now sliding at the point of contact with the dome.

To understand why this happens, we need to consider the forces acting on the sphere. The force of gravity is still acting on the sphere, causing it to accelerate down the dome. But now, there is also a frictional force acting on the sphere, which is responsible for the skidding motion.

The coefficient of static friction is a measure of the maximum frictional force that can be exerted between two surfaces without causing them to slide against each other. In this scenario, the coefficient of static friction between the sphere and the dome can be calculated by equating the maximum frictional force to the force of gravity acting on the sphere.

To do this, we can use the equation F = μN, where F is the maximum frictional force, μ is the coefficient of static friction, and N is the normal force between the sphere and the dome. The normal force is equal to the weight of the sphere, which is given by mg, where m is the mass of the sphere and g is the acceleration due to gravity.

Therefore, we can write the equation as μmg = mg sin 30, where μ is the coefficient of static friction we are trying to calculate. Solving for μ, we get μ = sin 30, which is equal to 0.5.

In conclusion, the coefficient of static friction between the sphere and the dome is 0.5. This means that the maximum frictional force that can be exerted between the two surfaces without causing them to slide against each other is half the weight of the sphere. I hope this explanation helps in understanding the concept of rotational dynamics