Question

In lab you assumed the pendulum rod (length 30.6 cm) was massless, with the cage (260 g) and ball (use 64 g) as a point mass at the end of the rod. Let’s repeat that calculation just so we are all using the same numbers. In lab, you related the final potential energy (use θ = 43 ° from vertical, where 0 ° corresponds to the initial (bottom) position) to the initial kinetic energy to get an initial velocity.

i. Find the final height of the mass.

ii. Find the change in potential energy.

iii. Set final potential energy equal to initial linear kinetic energy, and find the initial velocity.

iv. Convert this initial linear velocity directly to an initial angular velocity.

v. Find the moment of inertia of this system (about the pivot).

vi. Set final potential energy equal to initial rotational kinetic energy, and find the initial angular velocity. vii. Are the answers to (iv) and (vi) the same?

Answer 1

Hi,

Hope you are doing well.

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PART (i)

Lcose L-Lcoso

Given that,

Length of the rod, L = 30.6 m =0.306 m

Angle,

Ꮎ = Ꮞ3°

Therefore, Height reached by the mass:

h = L - L cos A = 0.306 m - (0.306 x cos 43°)

:.h=0.0822 m

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PART (ii)

Assume that the initial height=0 m. Then initial potential energy will be zero.

Mass of the bob (cage+ball), m= 260 g+64 g = 324 g = 0.324 kg

Therefore, Change in potential energy is given by:

AU = mgh = 0.324 kg x 9.8 m/s2 x 0.0822 m

AV = 0,261 )

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PART (iii)

According to Law of conservation of energy, Potential energy = kinetic energy.

AU = -mv

Therefore, Initial velocity is given by:

2AU 2 x 0.261 J V 0.324 kg

::V = 1.61123 m2/s2

:: V = 1.27 m/s

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PART (iv)

Angular velocity is given by:

W =- Linear velocity Radius

..w = 1.27 m/s 0,306 m

::w = 4.148 rad/s

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PART (v)

Moment of inertia of this system is given by:

I = m R = mL = 0.324 kg x (0.306 m)

:1= 0.030338 kg.m2

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PART (vi)

We have, Initial potential energy,

AV = 0,261 )

Rotational kinetic energy is given by,

Iw2 = AU

2 x 0.261 J V 0.030338 kg.m2

:.w= V17.206 rad/s

::w = 4.148 rad/s

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PART (vi)

YES, The values for angular velocity in Part (iv) and Part (vi) are same = 4.148 rad/s

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