Why Does Static Friction Cause Torque in a Rolling Ball?

[raisatantuico]

Homework Statement

a solid metal ball of mass M and radius R is rolling without slipping down and incline. A static friction force of magnitude f is acting on the ball. What is the magnitude of the torque due to static friction?
a.) zero b.) rfcos theta c.) rfsin theta d.)Rf e. MRF

Homework Equations

ki+ui=kf=uf (conservation of energy)

The Attempt at a Solution

since the ball is rolling without slipping, why is there friction involved? do we just simply ignore friction, so the answer will just be the definition of torque: rfsin theta ?

 

[Doc Al]

since the ball is rolling without slipping, why is there friction involved?

Without friction, the ball would slide not roll.

do we just simply ignore friction, so the answer will just be the definition of torque: rfsin theta ?

No, you cannot ignore friction. (And how is that answer ignoring friction anyway? It includes f!)

 

[raisatantuico]

Without friction, the ball would slide not roll.

No, you cannot ignore friction. (And how is that answer ignoring friction anyway? It includes f!)

f in the equation is force. torque= radius x force sin theta. net torque is also equal to angular acceleration x moment of inertia

where does friction come in this defintion of torque?

 

[Doc Al]

Here 'f' is the friction force. (What direction does friction act? What angle does it make with the radius?)

 

[rwisz]

I would like to clarify that even if the ball is rolling without slipping, there is still a static friction force acting on it. This is because the surface of the incline is exerting a normal force on the ball, causing the ball to experience a friction force in the opposite direction. This friction force is necessary to prevent the ball from slipping and to maintain its rolling motion.

Therefore, the magnitude of the torque due to static friction can be calculated using the equation τ = rfsinθ, where r is the radius of the ball, f is the magnitude of the static friction force, and θ is the angle between the radius vector and the direction of the friction force. This means that the correct answer is c.) rfsinθ.

Additionally, the conservation of energy equation mentioned in the problem can also be used to solve for the magnitude of the torque. Since the ball is rolling without slipping, there is no change in its kinetic energy, so the initial kinetic energy (due to its rolling motion) is equal to the final kinetic energy (due to its translational motion). This means that the initial kinetic energy can be expressed as 1/2Iω^2, where I is the moment of inertia of the ball and ω is its angular velocity. The final kinetic energy can be expressed as 1/2Mv^2, where M is the mass of the ball and v is its linear velocity. Equating these two expressions and solving for the angular velocity, we get ω = v/r. The torque can then be calculated as τ = Iα, where α is the angular acceleration of the ball. Since the ball is rolling without slipping, α = a/r, where a is the linear acceleration of the ball. Substituting these values into the torque equation, we get τ = Ia/r = Mra/r = Mrf, which is equivalent to c.) rfsinθ.

In conclusion, the correct answer is c.) rfsinθ, and the presence of friction in this scenario is necessary to maintain the ball's rolling motion without slipping.