A simple pendulum, 2.0 m in length, is released from rest when the support string is at an angle of 25 from the vertical. What is the speed of the suspended mass at the bottom of the swing?
Let A be the pendulum angle, which vaires sinusoidally with time. R is the pendulam length
R sin A = X
Xm = 2.0 sin 25 = 0.845 m is the oscillation amplitude (measured horizontally)
X = Xm sin (w t)
is the equation of motion
w is the angular frequency
w = sqrt(g/R) = 2.21 rad/s
Vmax = w*Xm = 2.21 rad/s*0.845 m
= 1.87 m/s
What is the mathmatical formula
To find the speed of the suspended mass at the bottom of the swing, we can use the equation for the period of a simple pendulum:
T = 2π√(L/g)
Where:
T is the period of the pendulum,
L is the length of the pendulum, and
g is the acceleration due to gravity.
First, let's convert the angle from degrees to radians:
25 degrees * (π/180) = 0.4363 radians
Next, we need to find the value of acceleration due to gravity, g. Assuming the standard value for g is approximately 9.8 m/s^2.
Now, we can calculate the period of the pendulum:
T = 2π√(2.0/g)
T = 2π√(2.0/9.8)
T ≈ 2π√(0.2040)
T ≈ 2π * 0.4519
T ≈ 2.837 seconds
The speed of the pendulum at the bottom of the swing is equal to the length of the pendulum (L) multiplied by the angular velocity (ω) at the bottom, which is equal to 2π divided by the period (T):
v = L * ω
v = 2.0 * (2π / 2.837)
v ≈ 2.0 * 2.216
v ≈ 4.432 m/s
Therefore, the speed of the suspended mass at the bottom of the swing is approximately 4.432 m/s.
To find the speed of the suspended mass at the bottom of the swing, we need to use the concept of conservation of mechanical energy. Mechanical energy is the sum of kinetic energy and potential energy.
Initially, when the pendulum is released from rest, it is at the maximum potential energy (at the highest point of the swing) and minimum kinetic energy.
At the bottom of the swing, the pendulum has maximum kinetic energy and minimum potential energy because it is at the lowest point of the swing.
The total mechanical energy remains constant throughout the motion of the pendulum because there is no external force acting on it to change its mechanical energy.
To find the speed at the bottom of the swing, we can equate the potential energy at the highest point to the kinetic energy at the lowest point.
The potential energy at the highest point is given by:
Potential Energy = m * g * h
Where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height from the lowest point to the highest point.
The kinetic energy at the lowest point is given by:
Kinetic Energy = 1/2 * m * v^2
Where m is the mass of the pendulum bob and v is the velocity of the pendulum bob at the lowest point.
Since the total mechanical energy is conserved, we can equate the potential energy and kinetic energy expressions:
m * g * h = 1/2 * m * v^2
We can cancel out the mass 'm' from both sides of the equation:
g * h = 1/2 * v^2
To find the velocity 'v', we just need to rearrange the equation:
v^2 = 2 * g * h
Taking the square root of both sides, we have:
v = √(2 * g * h)
Substituting the given values:
h = length of the pendulum = 2.0 m
g = acceleration due to gravity = 9.8 m/s²
v = √(2 * 9.8 * 2.0) = √(39.2) = 6.26 m/s
Therefore, the speed of the suspended mass at the bottom of the swing is 6.26 m/s.