A hemisphere of radius r is placed on a horizontal plane and a small mass of "m" is placed on the top of the hemisphere.
Find the height from the ground to the mass m,at the moment it loses it's contact with the hemisphere.And the contact is frictionless.
If we take the speed of the mass m, at the top of the hemisphere, v,the forces acting on the mass in that moment are mg,R and the centripetal force.
And the moment the mass loses its contact with the hemisphere,the forces acting on it are R and mg,and the centripetal force(which is zero as the speed is zero.)(because for the mass to lose its contact with the hemisphere, the speed should be zero as R≠0)
Hope my thoughts are correct and please correct me if they are wrong!
So do we have to solve this applying the law of conservation of energy? But how do find the speed v at the top as both R and v are unknowns.We can't simply equate R and mg because of the centripetal force acting on the mass in that moment!
Any help would be appreciated! Thank you!
And in both cases R cannot be the same too.
Try reaction=centripetal force,
mg(cos(θ))=mv^2/R
Yes, v^2 (at the time it leaves sphere) can be calculated using energy considerations.
But at the top if we apply F=ma towards the center of the hemisphere it would be,
mg-R=mv^2/r right? So how do we eliminate R?
You don't have to worry about the top, just assume that it was off balance by some random motion, and acceleration will start from there where KE=0, and PE=mg(R) if we take reference elevation at the centre of the sphere. Then at at certain θ<pi/2, mg(cos(θ))=mv^2/R
PE=mgR(1-cos(θ)=mv^2/2=KE
The R will cancel out when you substitute.
You're on the right track! To solve this problem, you can indeed apply the law of conservation of energy.
At the top of the hemisphere, the gravitational potential energy of the mass is converted into kinetic energy. The height from the ground to the mass at this point is the height of the hemisphere.
Now, let's consider the forces acting on the mass at the top of the hemisphere:
1. The weight of the mass: mg (directed vertically downward)
2. The normal force exerted by the hemisphere: R (perpendicular to the surface of the hemisphere)
3. The centripetal force: mv^2/r (directed towards the center of the hemisphere)
Since the contact is frictionless, the normal force provides the centripetal force required to keep the mass moving in a circular path on the hemisphere.
To find the speed v at the top of the hemisphere, you can use the conservation of energy. The initial potential energy at the height of the hemisphere is converted into kinetic energy at the top:
mgh = (1/2)mv^2
Where h is the height of the hemisphere.
Now, let's solve for v:
1. Cancel out the mass m from both sides of the equation.
2. Substitute the value of g (acceleration due to gravity, approximately 9.8 m/s^2).
3. Substitute the value of h (height of the hemisphere).
You will then have an equation only in terms of v. Solve for v to find the speed at the top of the hemisphere.
Once you've determined the speed at the top, you can use the forces at that point (R and mg) to find the height from the ground to the mass right before it loses contact with the hemisphere.