A pendulum bob has its maximum speed as 5m/s at lowest position o. Calculate the height of the Bob above o,where its velocity is zero.
1/2 m v^2 = m g h
h = v^2 / (2 g)
Vi=5.0m/s upwarrd
vf=0.0m/s
a=-9.8m/s/s
d=?
Vf2=Vi2+2ad
d=(Vf2-Vi2)÷2a
=25÷(2x9.8)=1.27m
To calculate the height of the pendulum bob where its velocity is zero, we can use the principle of conservation of mechanical energy.
The mechanical energy of a pendulum is the sum of its kinetic energy (KE) and potential energy (PE), given by the equation:
E = KE + PE
At the lowest position, the bob has its maximum speed of 5 m/s, which means all its energy is kinetic energy. Therefore, at the lowest position, the potential energy is zero (PE = 0) and the mechanical energy is equal to the kinetic energy:
E = KE
At the height where the velocity is zero, the bob has come to a stop, so its kinetic energy is zero (KE = 0). Thus, at this height, the mechanical energy is equal to the potential energy:
E = PE
Using the conservation of mechanical energy, we can equate the initial mechanical energy at the lowest position to the mechanical energy at the height where the velocity is zero:
E_initial = E_final
KE_initial + PE_initial = KE_final + PE_final
Since the bob's velocity is zero at the height in question, the kinetic energy at that height is zero:
0 + PE_initial = 0 + PE_final
Therefore, the potential energy at the lowest position is equal to the potential energy at the height where the velocity is zero:
PE_initial = PE_final
Now, let's consider the relationship between potential energy and height. The potential energy of an object near the surface of the Earth is given by the equation:
PE = m * g * h
where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the object.
Since the mass of the bob does not change, we can write the equation for potential energy as:
PE_initial = m * g * h_initial
PE_final = m * g * h_final
Now, equating the two potential energies, we get:
m * g * h_initial = m * g * h_final
The mass (m) and acceleration due to gravity (g) cancel out, leaving the equation:
h_initial = h_final
This shows that the height of the bob above the lowest position, where its velocity is zero, is the same as the height at the lowest position. In other words, the height remains constant throughout the bob's oscillation.