A pendulum bob is released from some initial height such that the speed of the bob at the bottom of the swing is 3.2 m/s.
The acceleration of gravity is 9.81 m/s
2. What is the initial height of the bob?
To determine the initial height of the bob, we can use the principle of conservation of energy. The total mechanical energy of the system is conserved, so we can equate the potential energy at the initial height to the kinetic energy at the bottom of the swing.
The potential energy of the bob at the initial height is given by the formula:
Potential Energy = m * g * h
Where:
m = mass of the bob
g = acceleration due to gravity
h = initial height
The kinetic energy of the bob at the bottom of the swing is given by the formula:
Kinetic Energy = (1/2) * m * v^2
Where:
m = mass of the bob
v = speed at the bottom of the swing
Since the total mechanical energy is conserved, we can equate the two energies:
m * g * h = (1/2) * m * v^2
We can cancel out the mass (m) on both sides of the equation:
g * h = (1/2) * v^2
Now, we can solve for the initial height (h):
h = (1/2) * (v^2) / g
Plugging in the given values:
v = 3.2 m/s
g = 9.81 m/s^2
h = (1/2) * (3.2^2) / 9.81
h = 0.518 meters
Therefore, the initial height of the bob is approximately 0.518 meters.
To find the initial height of the pendulum bob, we need to use the conservation of mechanical energy.
The mechanical energy of the bob is conserved as it swings back and forth. At the top of the swing, all of the energy is potential energy (PE), and at the bottom of the swing, all of the energy is kinetic energy (KE).
The formula for the conservation of mechanical energy is:
PE(initial) + KE(initial) = PE(final) + KE(final)
At the top of the swing, the kinetic energy is zero (KE(initial) = 0) since the bob is momentarily at rest. The potential energy is given by:
PE(initial) = mgh
Where m is the mass of the bob, g is the acceleration due to gravity (9.81 m/s²), and h is the initial height.
At the bottom of the swing, the potential energy is zero (PE(final) = 0) since the bob is at its lowest point. The kinetic energy is given by:
KE(final) = (1/2)mv²
Where v is the speed of the bob at the bottom of the swing.
Plugging in the values, the conservation equation becomes:
mgh + 0 = 0 + (1/2)mv²
Since the mass (m) cancels out, the equation simplifies to:
gh = (1/2)v²
Now, we can solve for h:
h = (1/2) v² / g
Plugging in the values given in the problem, we can calculate:
h = (1/2) (3.2 m/s)² / 9.81 m/s²
Simplifying this expression will give us the initial height of the bob.
mgh=1/2 mv^2
intial PE=final KE
solve for h.