To determine the speed of the pendulum at the lowest point of its path, we can use the principles of energy conservation.
1. Calculate the initial potential energy:
The potential energy (PE) of the pendulum at the starting height is given by the equation:
[tex]\[
PE = m \cdot g \cdot h
\][/tex]
where:
- [tex]\( m = 1.5 \, \text{kg} \)[/tex] (mass)
- [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex] (acceleration due to gravity)
- [tex]\( h = 0.4 \, \text{m} \)[/tex] (height)
Substituting the values:
[tex]\[
PE = 1.5 \cdot 9.8 \cdot 0.4
\][/tex]
The potential energy at the starting height is found to be:
[tex]\[
PE = 5.88 \, \text{J} \quad (\text{Joules})
\][/tex]
2. Energy conservation principle:
At the lowest point of the pendulum's path, all of the initial potential energy will have been converted into kinetic energy (KE), as there are no other forms of energy being considered (assuming no air resistance or friction).
The kinetic energy (KE) at the lowest point is given by:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
where [tex]\( v \)[/tex] is the velocity at the lowest point.
By the energy conservation principle,
[tex]\[
PE_{initial} = KE_{lowest}
\][/tex]
So,
[tex]\[
5.88 = \frac{1}{2} \cdot 1.5 \cdot v^2
\][/tex]
3. Solve for the velocity [tex]\( v \)[/tex]:
Rearrange the equation to solve for [tex]\( v \)[/tex]:
[tex]\[
5.88 = 0.75 v^2
\][/tex]
[tex]\[
v^2 = \frac{5.88}{0.75}
\][/tex]
[tex]\[
v^2 = 7.84
\][/tex]
[tex]\[
v = \sqrt{7.84}
\][/tex]
The velocity [tex]\( v \)[/tex] is:
[tex]\[
v \approx 2.80 \, \text{m/s}
\][/tex]
Thus, the speed of the pendulum when it reaches the lowest point of its path is approximately [tex]\(2.80 \, \text{m/s}\)[/tex].
