What am I Missing? Solving Conservation of Energy

[Vivek98phyboy]

Homework Statement

Below figure shows a small spherical ball of mass m rolling down the loop track. The ball is released on the linear portion at a vertical height H from the lowest point. The circular part shown has a radius R.

Find the tangential acceleration of the centre of mass of the ball when it is at A.

Relevant Equations

Started solving with Conservation of energy

mgH= mgR(1+sin(theta)) + ((Iw²)/2) + mv²/2

By solving conservation of energy, I was able to find the linear velocity which is
[10g(H-R-Rsin(theta))/7]^½ and by differentiating this with respect to "t", I arrived at the tangential acceleration value of -(5gcos(theta))/7 and found it to be in agreement with the solution provided in the book.

But when I draw a free body diagram of the ball by resolving it into radial and tangential component, If I had resolved it properly, i found that the Normal force and mg.sin(theta) points along the radius inwards and mgcos(theta) points along the tangential direction.

Given the scenario that mgcos(theta) is the only force along tangential direction, shouldn't the tangential acceleration be -gcos(theta)?

What am I missing?

 

[ehild]

Homework Statement:: Below figure shows a small spherical ball of mass m rolling down the loop track. The ball is released on the linear portion at a vertical height H from the lowest point. The circular part shown has a radius R.

Find the tangential acceleration of the centre of mass of the ball when it is at A.
Relevant Equations:: Started solving with Conservation of energy

mgH= mgR(1+sin(theta)) + ((Iw²)/2) + mv²/2

View attachment 266809
By solving conservation of energy, I was able to find the linear velocity which is
[10g(H-R-Rsin(theta))/7]^½ and by differentiating this with respect to "t", I arrived at the tangential acceleration value of -(5gcos(theta))/7 and found it to be in agreement with the solution provided in the book.

But when I draw a free body diagram of the ball by resolving it into radial and tangential component, If I had resolved it properly, i found that the Normal force and mg.sin(theta) points along the radius inwards and mgcos(theta) points along the tangential direction.

Given the scenario that mgcos(theta) is the only force along tangential direction, shouldn't the tangential acceleration be -gcos(theta)?

What am I missing?

You took rolling into account when calculating the tangential acceleration from the energy, but ignored rolling in the other case. it is also the static friction causing rolling that acts on the ball.

 

[wrobel]

An interesting thing appears at the point where the straight line conjugates with the circle: the acceleration jumps

 

[Delta2]

An interesting thing appears at the point where the straight line conjugates with the circle: the acceleration jumps

You mean because centripetal acceleration "suddenly" appears there? As far as i know acceleration need not be continuous, its velocity that is usually continuous.

 

[wrobel]

You mean because centripetal acceleration "suddenly" appears there?

As far as i know acceleration need not be continuous, its velocity that is usually continuous.

yes, but I believe that such things should be stressed

 

[Vivek98phyboy]

You took rolling into account when calculating the tangential acceleration from the energy, but ignored rolling in the other case. it is also the static friction causing rolling that acts on the ball.

You are right. When I took friction into account, I got it right.

Thank you

 

[Vivek98phyboy]

An interesting thing appears at the point where the straight line conjugates with the circle: the acceleration jumps

I didn't notice this. Thank you for pointing out this fact

 

[haruspex]

I didn't notice this. Thank you for pointing out this fact

Rate of change of acceleration is sometimes called 'jerk'. It is important in ergonomics. A standing passenger on a slowing bus has to brace herself against the inertial force; at the instant the bus stops the inertial force vanishes, causing her to lurch backwards.