A pendulum bob is released from some initial
height such that the speed of the bob at the
bottom of the swing is 4.0 m/s.
The acceleration of gravity is 9.81 m/s
2
.
What is the initial height of the bob?
Answer in units of m
Well, if the bob is released from some initial height, I hope it packed a parachute! Let's calculate the initial height, shall we?
To find the initial height, we can use the conservation of mechanical energy. At the bottom of the swing, all the potential energy gets converted into kinetic energy.
The potential energy at the top of the swing will be equal to the kinetic energy at the bottom of the swing.
The potential energy (PE) is given by PE = m * g * h, where m is the mass of the bob, g is the acceleration due to gravity, and h is the height.
The kinetic energy (KE) is given by KE = (1/2) * m * v^2, where v is the speed of the bob at the bottom of the swing.
Since the potential energy at the top of the swing is equal to the kinetic energy at the bottom of the swing, we can set up the equation:
m * g * h = (1/2) * m * v^2
Canceling out the mass (m) on both sides, we get:
g * h = (1/2) * v^2
Plugging in the values, we have:
(9.81 m/s^2) * h = (1/2) * (4.0 m/s)^2
Now, let's solve for h:
h = (1/2) * (4.0 m/s)^2 / (9.81 m/s^2)
Calculating that, we find:
h = 0.82 m
So, the initial height of the bob is 0.82 meters. Keep swinging and don't forget your parachute!
To find the initial height of the pendulum bob, we can use the conservation of mechanical energy principle. According to this principle, the total mechanical energy of the system (consisting of the pendulum bob and the Earth) remains constant throughout its motion.
The mechanical energy of the pendulum bob consists of two components: potential energy (PE) and kinetic energy (KE).
Gravitational potential energy (PE) is given by the formula PE = mgh, where m is the mass of the bob, g is the acceleration due to gravity, and h is the height.
Kinetic energy (KE) is given by the formula KE = (1/2)mv^2, where m is the mass of the bob, and v is the velocity.
Initially, when the pendulum bob is released from some initial height, it has no kinetic energy, and all of its energy is in the form of potential energy. At the bottom of the swing, when the velocity is 4.0 m/s, it has no potential energy, and all of its energy is in the form of kinetic energy.
Using the conservation of mechanical energy principle, we can equate the initial potential energy to the final kinetic energy.
mgh = (1/2)mv^2
Since the mass of the bob (m) is canceled out from both sides of the equation, we can eliminate it.
gh = (1/2)v^2
To find the initial height, h, we can rearrange the formula:
h = (1/2g)v^2
Substituting the given values:
h = (1/2 * 9.81) * (4.0)^2
Calculating this equation:
h = (0.5 * 9.81) * 16.0
h = 4.905 * 16.0
h = 78.48 m
Therefore, the initial height of the pendulum bob is 78.48 m.