A sphere of mass 50g and diameter 20cm is rolling with a speed of 5cm/sec . What will be its its total energy.
To determine the total energy of the rolling sphere, we need to consider both its kinetic energy and its potential energy. The total energy can be calculated as the sum of these two energies.
1. Kinetic Energy (KE):
The kinetic energy of a sphere is given by the equation:
KE = (1/2) * mass * velocity^2
Where:
- mass = 50g = 0.05 kg (convert grams to kilograms)
- velocity = 5 cm/sec = 0.05 m/sec (convert centimeters to meters)
Substituting these values into the equation:
KE = (1/2) * 0.05 kg * (0.05 m/sec)^2
= (1/2) * 0.05 kg * 0.0025 m^2/sec^2
= 0.0000625 kg * m^2/sec^2
2. Potential Energy (PE):
The potential energy of a rolling sphere can be calculated using the equation:
PE = (1/2) * mass * gravitational acceleration * height
Since the sphere is rolling, its height is equal to half of its diameter:
height = diameter/2 = 20 cm / 2 = 10 cm = 0.1 m (convert centimeters to meters)
The gravitational acceleration (g) is approximately 9.8 m/s^2.
Substituting these values into the equation:
PE = (1/2) * 0.05 kg * 9.8 m/s^2 * 0.1 m
= 0.000245 kg * m^2/s^2
3. Total Energy:
To get the total energy, we need to sum the kinetic energy and potential energy:
Total Energy = KE + PE
= 0.0000625 kg * m^2/sec^2 + 0.000245 kg * m^2/s^2
= 0.0003075 kg * m^2/s^2
Therefore, the total energy of the rolling sphere is approximately 0.0003075 kg * m^2/s^2.
To calculate the total energy of the rolling sphere, we need to consider both its kinetic energy and potential energy.
1. Kinetic Energy (KE):
The kinetic energy of a rolling sphere is given by the formula:
KE = (1/2) * I * ω²
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
Since the sphere is rolling without slipping, we need to consider the rolling motion using the relationship between linear velocity and angular velocity:
v = ω * r
where v is the linear velocity, ω is the angular velocity, and r is the radius of the sphere.
Given:
- Mass (m): 50 g = 0.05 kg
- Diameter (d): 20 cm = 0.2 m
- Speed (v): 5 cm/sec = 0.05 m/sec
First, we need to find the radius (r) of the sphere, which is half of the diameter:
r = d/2 = 0.2/2 = 0.1 m
Now, let's calculate the angular velocity (ω) using:
v = ω * r
ω = v / r = 0.05 / 0.1 = 0.5 rad/sec
Next, we need to calculate the moment of inertia (I) for a solid sphere:
I = (2/5) * m * r²
I = (2/5) * 0.05 * (0.1)² = 0.002 kg m²
Now, we can calculate the kinetic energy (KE):
KE = (1/2) * I * ω²
KE = (1/2) * 0.002 * (0.5)²
KE = 0.00025 J
2. Potential Energy (PE):
The potential energy of a rolling sphere is given by the formula:
PE = m * g * h
where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height.
Since the sphere is rolling, it will have no potential energy if we consider the ground level as the reference point. Therefore, PE is zero.
3. Total Energy:
The total energy (E) of the rolling sphere is the sum of its kinetic energy and potential energy:
E = KE + PE
E = 0.00025 + 0
E = 0.00025 J
Therefore, the total energy of the rolling sphere is 0.00025 Joules.