A small solid sphere, with radius 0.25 cm and mass 0.61 g rolls without slipping on the inside of a large fixed hemisphere with radius 17 cm and a vertical axis of symmetry. The sphere starts at the top from rest.
(a) What is its kinetic energy at the bottom?
PE = 0.0061 kg * g * 0.017 m = 0.010 J
KE = PE = 0.010 J
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Haven't you made mistakes in two of those unit conversions? 0.61g = 0.00061kg not 0.0061kg, and 17cm = 0.17m not 0.017m.
To find the kinetic energy of the small solid sphere at the bottom, we need to consider the conservation of energy.
The total mechanical energy of the system (small solid sphere + large fixed hemisphere) will remain constant throughout the motion. Since the sphere starts from rest at the top, it will convert its potential energy at the top into kinetic energy as it rolls down to the bottom.
To calculate the kinetic energy, we need to first find the potential energy at the top and then subtract it from the total mechanical energy.
1. Calculate the potential energy at the top:
The potential energy of an object at a certain height can be given by the equation: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
- Mass of the sphere (m) = 0.61 g = 0.00061 kg
- Height at the top = radius of the large fixed hemisphere = 17 cm = 0.17 m
- Acceleration due to gravity (g) = 9.8 m/s^2
PE_top = m * g * h
PE_top = 0.00061 kg * 9.8 m/s^2 * 0.17 m
PE_top = 0.00098912 Joules
2. Calculate the total mechanical energy at the top:
Since the sphere starts from rest at the top, its initial kinetic energy is zero. Therefore, at the top, the total mechanical energy is equal to the potential energy.
Total mechanical energy_top = PE_top
Total mechanical energy_top = 0.00098912 Joules
3. Calculate the kinetic energy at the bottom:
As the sphere rolls without slipping, it will convert its potential energy at the top into kinetic energy at the bottom.
Total mechanical energy_top = Kinetic energy_bottom + Potential energy_bottom
Since the sphere has reached the bottom, its potential energy at the bottom is zero (h=0). Therefore, we can rewrite the equation as:
Total mechanical energy_top = Kinetic energy_bottom + 0
Rearranging the equation:
Kinetic energy_bottom = Total mechanical energy_top
So, the kinetic energy at the bottom is equal to the total mechanical energy at the top.
Kinetic energy_bottom = Total mechanical energy_top
Kinetic energy_bottom = 0.00098912 Joules
Therefore, the kinetic energy of the small solid sphere at the bottom is approximately 0.00098912 Joules.