A pendulum bob has its speed as 3m/s at the lowest position 0. Calculate the height of the bob above 0 where its velocity is zero
To calculate the height of the bob above the lowest position where its velocity is zero, we can use the conservation of mechanical energy.
The mechanical energy of a pendulum bob consists of two parts: potential energy (PE) and kinetic energy (KE).
At the lowest position, all the energy is in the form of kinetic energy, and as the bob rises, it converts its kinetic energy to potential energy. At the highest point, all the energy is in the form of potential energy, and as the bob falls back down, it converts its potential energy to kinetic energy.
At the lowest position, the speed of the bob is given as 3 m/s, which corresponds to its kinetic energy.
KE = 1/2 * m * v^2
where m is the mass of the bob and v is its velocity.
Since the velocity is given as 3 m/s, we can substitute the values into the equation:
KE = 1/2 * m * (3)^2
= 1/2 * m * 9
= 4.5 * m
Now, at the highest point, the velocity is zero, which means all the energy is in the form of potential energy.
PE = m * g * h
where g is the acceleration due to gravity and h is the height above the lowest position.
Since the mechanical energy is conserved, we can equate the kinetic energy at the lowest position to the potential energy at the highest position:
KE = PE
4.5 * m = m * g * h
The mass (m) cancels out from both sides, so we're left with:
4.5 = g * h
Now, we need to know the value of the acceleration due to gravity (g). On Earth, the standard value for g is approximately 9.8 m/s^2.
Substituting g = 9.8 m/s^2 into the equation:
4.5 = 9.8 * h
To solve for h, we can rearrange the equation:
h = 4.5 / 9.8
h ≈ 0.459 m
Therefore, the height above the lowest position where the velocity is zero is approximately 0.459 meters.
To calculate the height of the bob above the reference point (0) where its velocity is zero, we can use the principles of conservation of mechanical energy.
1. Identify the information given:
- Initial speed of the pendulum bob (v_i) = 3 m/s
- Final velocity of the pendulum bob (v_f) = 0 m/s
2. Understand the concept:
- The pendulum's bob undergoes simple harmonic motion, where the total mechanical energy remains constant.
- Mechanical energy (E) is the sum of kinetic energy (KE) and potential energy (PE): E = KE + PE.
- At the highest point (maximum height), all the energy is in the form of potential energy, and the kinetic energy is zero.
- At the lowest point (0 height), all the energy is in the form of kinetic energy, and the potential energy is zero.
- The mechanical energy remains constant throughout the motion: E_initial = E_final.
3. Apply conservation of mechanical energy to solve the problem:
- E_initial = E_final
- KE_initial + PE_initial = KE_final + PE_final
At the highest point:
- KE_initial = 0 (kinetic energy is zero)
- PE_initial = mgh (potential energy is in the form of gravitational potential energy)
At the lowest point (0 height):
- KE_final = 1/2 * mv^2 (kinetic energy is in the form of translational kinetic energy)
- PE_final = 0 (potential energy is zero)
This leads to the equation:
0 + mgh = 1/2 * mv^2 + 0
4. Substitute the given values:
0 + mgh = 1/2 * m * (3)^2 + 0
Simplify the equation:
gh = 1/2 * 3^2
Solve for h:
h = (1/2 * 9) / g
h = 4.5 / g
The height of the pendulum bob above the reference point (0) where its velocity is zero is given by h = 4.5 / g, where g is the acceleration due to gravity.
it reaches maximum height (velocity=0) when the KE at the bottom has been converted to PE. So,
1/2 mv^2 = mgh
9/2 = 9.8h
now finish it off