A ball, of mass 0.15 Kg, is held on the inside of a smooth hemispherical bowl at a height 0.16 m above the base. It is then released. Assuming there is no friction, the acceleration due to gravity, g, is 10 m/s2 and the potential energy is zero at the base of the bowl - What is the maximum value of kinetic energy?
Maximum kinetic energy equals initial potential energy.
To find the maximum value of kinetic energy, we need to first find the speed of the ball when it reaches the bottom of the bowl. We can do this by using the principle of conservation of energy.
The total mechanical energy of the system remains constant throughout the motion. Initially, the ball has only gravitational potential energy, and at the bottom of the bowl, it has only kinetic energy.
The formula for gravitational potential energy is given by:
Potential Energy = mass × gravity × height
Given:
Mass of the ball (m) = 0.15 kg
Height of the ball above the base (h) = 0.16 m
Acceleration due to gravity (g) = 10 m/s²
Potential Energy at the top = mass × gravity × height
Potential Energy at the top = 0.15 kg × 10 m/s² × 0.16 m
Now, since the potential energy at the bottom is zero (as given), we can equate the potential energy at the top to the kinetic energy at the bottom:
Potential Energy at the top = Kinetic Energy at the bottom
0.15 kg × 10 m/s² × 0.16 m = (1/2) × mass × (velocity)²
Simplifying the equation:
0.15 kg × 10 m/s² × 0.16 m = (1/2) × 0.15 kg × (velocity)²
Now, solve for the velocity:
Velocity² = (2 × 0.15 kg × 10 m/s² × 0.16 m) / 0.15 kg
Velocity = √(2 × 10 m/s² × 0.16 m)
Calculating the velocity:
Velocity = √(3.2 m²/s²)
Velocity = 1.79 m/s
The maximum value of kinetic energy occurs when the ball reaches the bottom of the bowl. Therefore, the maximum kinetic energy is given by:
Kinetic Energy = (1/2) × mass × (velocity)²
Plugging in the values:
Kinetic Energy = (1/2) × 0.15 kg × (1.79 m/s)²
Calculating the kinetic energy:
Kinetic Energy = 0.0806 Joules
Therefore, the maximum value of kinetic energy is 0.0806 Joules.
To find the maximum value of kinetic energy, we need to determine the height at which the ball will have maximum speed, and then calculate the kinetic energy at that point.
Given:
Mass of the ball (m) = 0.15 kg
Height above the base (h) = 0.16 m
Acceleration due to gravity (g) = 10 m/s^2
The potential energy of an object at a certain height is given by the equation:
PE = mgh
Where m is the mass, g is the acceleration due to gravity, and h is the height.
At the base of the bowl, the potential energy is zero. Therefore, the initial potential energy (PEi) of the ball is zero.
At the maximum height, the potential energy (PEf) is given by:
PEf = mgh
The potential energy at the base (PEi) is equal to the kinetic energy at the maximum height (KEf), so:
KEf = PEi = 0
At the maximum height, all the potential energy is converted into kinetic energy. Therefore:
KEf = PEf = mgh
Substituting the given values into the equation, we have:
KEf = mgh
= (0.15 kg)(10 m/s^2)(0.16 m)
= 0.24 J
Therefore, the maximum value of kinetic energy is 0.24 Joules.