ExamKRACKERS 7th edition physics Problem #36

[dahmsom]

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Hello,
I know that the acceleration of the ball will be a= gsin theta. however i did a TBR problem like this, and it says that the acceleration will increase and velocity increase, but the answer key for examkracker says the acceleration and velocity will decrease because the theta decrease. Can anyone clarify this?

 

[MrRed]

Alright, this is how I would reason it.
So the only forces acting on the ball would be 1. gravity 2. friction 3. air resistance.
So we can definitely ignore air resistance, and I'm assuming we should consider friction.
So at the top there where the ball starts from, its total energy would be E = mgh + 1/2mv^2
At rest it has no kinetic energy so all of it's energy is gravitational potential: mgh.

As it rolls, the angle (theta) of its descent lessens, thus the kinetic friction on the ball will also increase (friction constant x normal force).
More friction means robbing the ball itself of its kinetic energy at a faster rate. At first, when the ball is at the very top, the angle of its descent is very near 90 degrees. This means the friction force on it is essentially zero. As it rolls and the frictional force becomes more significant, the energy of the ball from its relative position to the Earth (mgh) that has been converted into kinetic energy (1/2mv^2) is taken away faster and faster.
So how does this happen? Well you don't change its mass, so the only way that it's energy can decrease is by slowing the ball, decreasing velocity.

Since Force = m x a and work = F x d and the change in kinetic energy is directly equivalent to the work done on the ball in this case, the acceleration must also decrease.
This is what I think -- I'm not 100% sure if it is correct, but it does agree with ExamKrackers, so I don't know.

 

[edgerock24]

Simple/stupid reasoning I would have regarding this Q:

The question stem says "as the ball rolls down." This means we don't assume an initial velocity of zero. The ball is already rolling in our case -- it already has a velocity of some sort. Now, as it rolls, friction repeatedly acts upon the ball. This slows it down, and will eventually stop the ball from moving. So, our velocity is decreasing.

Next, for acceleration: we are only talking about vertical acceleration. This vertical acceleration is only coming from the acceleration due to gravity (g). The ball will roll down the slope, and at first, really feel the acceleration due to gravity. The further it moves down the slope, the less the acceleration due to gravity will have an effect. Eventually, the ball will only be moving horizontally. The acceleration due to gravity doesn't effect horizontal motion. In other words, as this objects motion transfers from vertical to horizontal, the more its vertical acceleration (g) will decrease.

Hope that sorta helps.

 

[dahmsom]

Thanks @edgerock24 wow I didn't read it carefully. Thanks for the response