A frictionless pendulum is made with a bob of mass 19.7 kg. The bob is held at height = 0.934 meter above the bottom of its trajectory, and then pushed forward with an initial speed of 2.93 m/s. What amount of mechanical energy does the bob have when it reaches the bottom?
According to the law of conservation of energy in frictionless case the total mechanical energy of the pendulum is constant
E = mgh +mv²/2 = 19.7•9.8•0.934 + 19.7•2.93²/2 = …
therefore the answer is 265 J
To find the amount of mechanical energy the bob has when it reaches the bottom, we need to consider the initial potential energy and kinetic energy of the system.
1. Calculate the initial potential energy:
The potential energy of the bob at the initial height can be calculated using the formula:
Potential energy = mass * acceleration due to gravity * height
Potential energy = 19.7 kg * 9.8 m/s^2 * 0.934 meters
2. Calculate the initial kinetic energy:
The kinetic energy of the bob at the initial speed can be calculated using the formula:
Kinetic energy = 0.5 * mass * velocity^2
Kinetic energy = 0.5 * 19.7 kg * (2.93 m/s)^2
3. Sum the initial potential energy and kinetic energy to get the total mechanical energy:
Total mechanical energy = Initial potential energy + Initial kinetic energy
Calculate each part of the equation to find the answer.
To find the amount of mechanical energy the bob has when it reaches the bottom, we need to consider the initial potential energy and the initial kinetic energy of the bob, and compare it to the potential energy and kinetic energy at the bottom.
First, let's find the initial potential energy (PE_initial) and initial kinetic energy (KE_initial) of the bob.
The initial potential energy is given by the formula:
PE_initial = m * g * h
where m is the mass of the bob, g is the acceleration due to gravity, and h is the height.
In this case, the mass of the bob (m) is 19.7 kg, the acceleration due to gravity (g) is approximately 9.8 m/s^2, and the height (h) is 0.934 meter.
So, the initial potential energy is:
PE_initial = 19.7 kg * 9.8 m/s^2 * 0.934 meter
Next, let's find the initial kinetic energy using the formula:
KE_initial = 0.5 * m * v^2
where m is the mass of the bob, and v is the initial velocity.
In this case, the mass of the bob (m) is 19.7 kg, and the initial velocity (v) is 2.93 m/s.
So, the initial kinetic energy is:
KE_initial = 0.5 * 19.7 kg * (2.93 m/s)^2
Now, let's consider the potential energy (PE_bottom) and kinetic energy (KE_bottom) at the bottom of the trajectory.
At the bottom, the height (h_bottom) is 0 meter, so the potential energy is zero:
PE_bottom = 0
The kinetic energy at the bottom is given by the formula:
KE_bottom = 0.5 * m * v^2
where m is the mass of the bob, and v is the velocity at the bottom.
Since the pendulum swings freely without friction, the initial velocity (2.93 m/s) is the same as the velocity at the bottom.
So, the kinetic energy at the bottom is:
KE_bottom = 0.5 * 19.7 kg * (2.93 m/s)^2
Now, we can calculate the total mechanical energy at the bottom by adding the potential energy and kinetic energy at the bottom:
Mechanical energy at the bottom (E_bottom) = PE_bottom + KE_bottom = 0 + 0.5 * 19.7 kg * (2.93 m/s)^2
Now, let's calculate the value of E_bottom.