A simple pendulum of length 2.00m is made with a mass of 2.00kg. The mass has speed of 3.00 m/s when the pendulum is 30 degrees above its lowest position.
-What is the maximum angle away from the lowest position the pendulum will reach?
-What is the speed of the mass when the pendulum is 45 degrees above its lowest position?
Work done: H = 2 - 2(cos30)
= .2679491924
mgh(max)= mgh + 1/2(mv^2)
To find the maximum angle away from the lowest position, we can use the principle of conservation of mechanical energy. The total mechanical energy of the pendulum is conserved, which means the sum of its potential energy and kinetic energy remains constant throughout the motion.
Given:
Length of the pendulum, L = 2.00 m
Mass of the pendulum, m = 2.00 kg
Initial speed of the mass, v = 3.00 m/s
First, we need to find the height at the maximum angle (height, h(max)) using the equation: h = L - L(cosθ), where θ is the angle above the lowest position.
For the given value of 30 degrees:
h(max) = 2 - 2(cos30)
= 2 - 2(√3/2)
= 2 - √3
Now, let's find the maximum potential energy, PEmax, at the highest position. The potential energy at the highest position is given by the equation: PEmax = mgh(max)
PEmax = (2.00 kg)(9.8 m/s^2)(2 - √3)
Next, let's find the maximum kinetic energy, KEmax, using the equation: KEmax = 1/2 mv^2
KEmax = 1/2 (2.00 kg)(3.00 m/s)^2
Since the total mechanical energy is conserved, it can be expressed as the sum of potential and kinetic energy:
Total mechanical energy (Emech) = PEmax + KEmax
Now, using the principle of conservation of mechanical energy, we can find the maximum angle away from the lowest position (θ(max)):
Emech = PEmax + KEmax
θ(max) = arccos((h(max) - Emech) / (2(L - 1/2 mv^2)))
To find the speed of the mass when the pendulum is 45 degrees above its lowest position, we can use the conservation of mechanical energy again. We will calculate the height at 45 degrees and then use the equation for total mechanical energy to find the speed.
Repeat the same steps as mentioned above to find the height at 45 degrees (h(45)).
Now, using the equation: Emech = PEmax + KEmax, we can find the kinetic energy at 45 degrees, KE(45).
Next, we can use the equation for kinetic energy, KEmax = 1/2 mv^2, to find the velocity or speed of the mass at 45 degrees (v(45)).
Therefore, to summarize:
1. To find the maximum angle away from the lowest position:
- Calculate h(max) using h = L - L(cosθ).
- Find the potential energy at the highest position, PEmax = mgh(max).
- Find the kinetic energy at the highest position, KEmax = 1/2 mv^2.
- Use Emech = PEmax + KEmax to find θ(max).
2. To find the speed of the mass when the pendulum is 45 degrees above its lowest position:
- Calculate h(45) using h = L - L(cosθ).
- Find the potential energy at 45 degrees, PE(45) = mgh(45).
- Find the kinetic energy at 45 degrees, KE(45) = Emech - PE(45).
- Use KE(45) = 1/2 mv^2 to find v(45).
Remember to plug in the given values and perform the necessary calculations to obtain the final answers.
To find the maximum angle away from the lowest position, we can use the conservation of mechanical energy. The potential energy at the highest point (when the pendulum is at the maximum angle) will be equal to the initial energy of the pendulum.
1. Calculate the potential energy at the highest point:
Potential energy (mgh) = mass (m) × acceleration due to gravity (g) × height (h) = 2.00 kg × 9.8 m/s^2 × (.2679491924 m) = 5.242 Joules
2. Calculate the kinetic energy at the initial position:
Initial kinetic energy = 1/2 × mass (m) × velocity (v)^2 = 1/2 × 2.00 kg × (3.00 m/s)^2 = 9.00 Joules
3. Apply the conservation of energy equation:
Potential energy at highest point = Initial kinetic energy
5.242 Joules = 9.00 Joules
Since the potential energy at the highest point is less than the initial kinetic energy, the pendulum will not reach the maximum angle. The maximum angle away from the lowest position will be less than 30 degrees (the initial angle).
To find the speed of the mass when the pendulum is 45 degrees above its lowest position, we can use the conservation of mechanical energy again.
1. Calculate the potential energy at the given angle:
Potential energy (mgh) = 2.00 kg × 9.8 m/s^2 × [2 - 2(cos 45 degrees)] = 9.365 Joules
2. Apply the conservation of energy equation:
Potential energy at the given angle = Final kinetic energy
9.365 Joules = 1/2 × mass (m) × final velocity (v)^2
Since we have the mass (2.00 kg), we can solve the equation for the final velocity.
9.365 Joules = 1/2 × 2.00 kg × (final velocity)^2
Divide both sides by 1 kg:
4.6825 Joules = (final velocity)^2
Take the square root of both sides:
final velocity = √(4.6825 Joules) = 2.164 m/s
Therefore, the speed of the mass when the pendulum is 45 degrees above its lowest position is approximately 2.164 m/s.
m g h + (1/2)m v^2 = total energy = constant
h is height above hanging straight down
mg(2-2 cos 30)+.5m*9 = total energy
= m g (2 -cos max angle when v = 0)
notice m has nothing to do with this problem, cancels
= mg(2-cos45) +.5 m v^2