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In the figure below, a solid brass ball of mass m and radius r will roll without slipping along the loop-the-loop track when released from rest along the straight section. For the following answers use g for the acceleration due to gravity, and m, r, and R, as appropriate, where all quantities are in SI units.

(a) From what minimum height h above the bottom of the track must the marble be released to ensure that it does not leave the track at the top of the loop? (The radius of the loop-the-loop is R. Assume R r.)

(b) If the marble is released from height 6R above the bottom of the track, what is the magnitude of the horizontal component of the force acting on it at point Q?

Attempt

a)we know that total energy at the height h is potential energy when the body is at rest then

total energy at height h is mgh

Total energy at the bottom of the loop is (1/2)mu2

then mgh = (1/2)mu2

Now total energy the top of the loop is (1/2)mv2 +mg(2R)

Then mgh = (1/2)mv2 +mg(2R)

given that ball is on the verge of leaving the track when it reaches the top of the loop

then mg = mv2/R

then mgh = (1/2)(mgR) +mg(2R)

h = (R/2) + 2R

h = 5R/2

b) Now given that h = 6R

then total energy at height h is mgh

Total energy at the bottom of the loop is (1/2)mu2

then mgh = (1/2)mu2

then mg*6R = (1/2)mu2

Now total energy the point Q is (1/2)mv2 +mg(R)

Then (1/2)mu2 = (1/2)mv2 +mg(R)

mg*6R = (1/2)mv2 +mg(R)

Then 5mgR = (1/2)mv2

then 10mgR = mv2

Now total horizontal force on the ball at point Q is mv2/R = 10mgR/R = 10mg

But the system is saying my answers is wrong, what am i doing wrong?

(a) From what minimum height h above the bottom of the track must the marble be released to ensure that it does not leave the track at the top of the loop? (The radius of the loop-the-loop is R. Assume R r.)

(b) If the marble is released from height 6R above the bottom of the track, what is the magnitude of the horizontal component of the force acting on it at point Q?

Attempt

a)we know that total energy at the height h is potential energy when the body is at rest then

total energy at height h is mgh

Total energy at the bottom of the loop is (1/2)mu2

then mgh = (1/2)mu2

Now total energy the top of the loop is (1/2)mv2 +mg(2R)

Then mgh = (1/2)mv2 +mg(2R)

given that ball is on the verge of leaving the track when it reaches the top of the loop

then mg = mv2/R

then mgh = (1/2)(mgR) +mg(2R)

h = (R/2) + 2R

h = 5R/2

b) Now given that h = 6R

then total energy at height h is mgh

Total energy at the bottom of the loop is (1/2)mu2

then mgh = (1/2)mu2

then mg*6R = (1/2)mu2

Now total energy the point Q is (1/2)mv2 +mg(R)

Then (1/2)mu2 = (1/2)mv2 +mg(R)

mg*6R = (1/2)mv2 +mg(R)

Then 5mgR = (1/2)mv2

then 10mgR = mv2

Now total horizontal force on the ball at point Q is mv2/R = 10mgR/R = 10mg

But the system is saying my answers is wrong, what am i doing wrong?