A simple pendulum has a length of 1.5 m and is pulled a distance y = 0.24 m to one side and then released. (Assume small angle.)
What is the speed of the pendulum when it passes through the lowest point on its trajectory?
Is this horizontal distance? If so, draw the triangle. 1.5 down, .24 left, and hypotenuse is sqrt (1.5^2+.24^2)
but the pendulum is not that long, it is 1.5, so at what height is the bob at?
similar triangles: h/1.5=(sqrt(1.5^2+.24^2) -1.5)/1.5
solve for h.
then mgh=1/2 m v^2 solve for v.
0.499
To determine the speed of the pendulum at its lowest point, you can use the principles of conservation of energy. The potential energy at the highest point of the pendulum is converted into kinetic energy at the lowest point.
To calculate the speed, you first need to find the potential energy at the highest point. The potential energy of a simple pendulum is given by the formula:
PE = mgh
where PE is the potential energy, m is the mass of the pendulum (assumed to be a point mass), g is the acceleration due to gravity, and h is the height.
In this case, since the pendulum is pulled sideways, we need to find the vertical height at the highest point. The height can be calculated using the cosine of the initial displacement angle (since it's a small angle):
h = L - L * cos(theta)
where L is the length of the pendulum and theta is the initial displacement angle.
Now we have all the information to calculate the potential energy at the highest point:
PE = mgh = mg(L - L * cos(theta))
Next, we can equate the potential energy at the highest point to the kinetic energy at the lowest point:
1/2 mv^2 = PE
Solving for v (the speed), we get:
v = √(2gh(1 - cos(theta)))
Substituting the given values into the equation:
- Length of the pendulum (L) = 1.5 m
- Displacement (y) = 0.24 m
- Gravitational acceleration (g) = 9.8 m/s^2
First, calculate the initial displacement angle (theta) using the equation:
theta = arcsin(y / L)
Then, calculate the speed (v) using the equation:
v = √(2gh(1 - cos(theta)))
Plug in the values and you can calculate the speed of the pendulum at its lowest point.