Non slip ball down an incline of theta

[mikegovikes]

non slip ball down an incline of theta!

Homework Statement

A uniform solid sphere rolls down an incline without slipping. If the linear acceleration of the center of mass of the sphere is 0.16g, then what is the angle the incline makes with the horizontal?

Homework Equations

F=ma T=I(alpha) I=(2/5)MR^2 a=r(alpha)

The Attempt at a Solution

I know that you have to use the first two equations. I made a free body diagram of the bowling ball, but i am confused about the first equation F=ma. is F the force of kinetic friction? how am i supposed to solve for the kinetic friction? is it equal to Mgsin theta where M is the mass of the bowling ball? then there would not be acceleration. I am confused

 

[ehild]

When a body rolls the point where it touches the ground is in rest with respect to the ground. It is static friction you need to use, and it acts at the rim of the sphere. But gravity acts too, at the centre of the sphere and its magnitude is Mg sin(theta).
So you have two forces, but there is also their torque.

ehild

 

[eczeno]

hello,

Look at the torques about the point where the sphere makes contact with the incline. in that case you have the wrong moment of inertia; you need to use the parallel axis theorem (you should get I = (7/5)MR^2). everything works out for me this way.

cheers

oh, also, it is static friction, not kinetic friction, when there is rolling without slipping.

 

[SammyS]

For F = ma, you need to use the Net force.

 

[rwisz]

.
I would approach this problem by first clarifying the given information and identifying the unknowns. The given information states that a solid sphere is rolling down an incline without slipping and has a linear acceleration of 0.16g. The unknown in this problem is the angle of the incline, theta.

Next, I would use the equations provided to analyze the forces and motion of the sphere. The first equation, F=ma, represents Newton's Second Law and states that the net force on an object is equal to its mass times its acceleration. In this case, the net force on the sphere is due to the force of gravity, which is equal to the mass of the sphere times the acceleration due to gravity, g, and the force of kinetic friction, which is equal to the coefficient of kinetic friction times the normal force. The normal force is equal to the weight of the sphere, which is mg, multiplied by the cosine of the angle of the incline.

The second equation, T=I(alpha), represents the relationship between torque, moment of inertia, and angular acceleration. In this case, the torque on the sphere is due to the force of kinetic friction, which is acting at a distance of the radius of the sphere from its axis of rotation. The moment of inertia of a solid sphere is equal to (2/5)MR^2, where M is the mass of the sphere and R is the radius.

Using these equations and setting the net force equal to the torque, we can solve for the angle of the incline. This will involve some algebraic manipulation and substitution of known values. Once we have solved for theta, we can verify our solution by plugging it back into the equations and ensuring that the given linear acceleration is consistent with the calculated values.

In conclusion, as a scientist, I would approach this problem by carefully analyzing the given information, using relevant equations to model the forces and motion of the sphere, and solving for the unknown angle of the incline.