0.5 * 50(1-cos35°)
It starts out as PE, and is converted to KE at the bottom of the swing.
It starts out as PE, and is converted to KE at the bottom of the swing.
First, let's find the vertical height of the mass, which will be the maximum potential energy. To do that, we need to find the vertical component of the string length. Since the angle is 35 degrees, the vertical component will be 50 cm * sin(35 degrees).
Now, let's compute the maximum potential energy by multiplying the mass (500 gm) by the vertical height (50 cm * sin(35 degrees)). Don't forget to convert grams to kilograms!
As for the kinetic energy, when the mass is at its highest point, it momentarily stops before falling back down. Therefore, the kinetic energy at the maximum height is zero. But fear not, the kinetic energy will increase as the mass swings down!
So, to be precise, the maximum potential energy will be [500 gm * 9.8 m/s^2 * (50 cm * sin(35 degrees))]/1000 (to convert grams to kilograms).
To summarize, the maximum potential energy is [500 gm * 9.8 m/s^2 * (50 cm * sin(35 degrees))]/1000, and the kinetic energy is zero at the highest point. Enjoy swinging into those energy calculations!
First, let's calculate the height of the pendulum's mass at its maximum displacement. We can use the sine function:
sin(theta) = opposite / hypotenuse
sin(35 degrees) = h / 50 cm
h = 50 cm * sin(35 degrees)
h ≈ 50 cm * 0.5736
h ≈ 28.68 cm
Now, let's convert the height to meters:
h = 28.68 cm * (1 m / 100 cm)
h = 0.2868 m
The maximum potential energy of the pendulum can be calculated using the formula:
Potential Energy = mass * gravitational acceleration * height
m = 500 g * (1 kg / 1000 g)
m = 0.5 kg
g = 9.8 m/s^2
Potential Energy = 0.5 kg * 9.8 m/s^2 * 0.2868 m
Potential Energy ≈ 1.408 Joules
The maximum kinetic energy of the pendulum occurs when the mass is at the lowest point of its swing, which is when it has maximum velocity. At this point, all potential energy is converted into kinetic energy.
Therefore, the maximum kinetic energy of the pendulum will be equal to the maximum potential energy:
Kinetic Energy ≈ 1.408 Joules
So, both the maximum potential energy and kinetic energy of the pendulum are approximately 1.408 Joules.
To solve the problem, we'll need the mass of the pendulum (500 grams or 0.5 kg) and the length of the string (50 cm or 0.5 meters).
1. Maximum Potential Energy:
The potential energy of an object in a gravitational field is given by the formula:
Potential Energy (PE) = mass (m) × acceleration due to gravity (g) × height (h)
In this case, the height (h) is the vertical distance between the rest position and the highest point reached by the pendulum.
But we don't have the height directly; we have an angle (θ) between the vertical and the string. So, we can use trigonometry to find the height.
Using the given angle (θ = 35 degrees) and the length of the string (L = 0.5 meters):
Height (h) = L × sin(θ)
Height (h) = 0.5 × sin(35)
Height (h) ≈ 0.287 meters
Now we can calculate the potential energy using the mass (m = 0.5 kg), acceleration due to gravity (g ≈ 9.8 m/s^2), and the height (h ≈ 0.287 meters):
Potential Energy (PE) = m × g × h
Potential Energy (PE) ≈ 0.5 × 9.8 × 0.287
Potential Energy (PE) ≈ 1.412 Joules
Therefore, the maximum potential energy of the pendulum is approximately 1.412 Joules.
2. Maximum Kinetic Energy:
The kinetic energy of an object is given by the formula:
Kinetic Energy (KE) = 0.5 × mass (m) × velocity (v)^2
At the lowest point of the swing, the pendulum has converted all potential energy into kinetic energy, so the gravitational potential energy is zero.
To find the velocity (v) at the lowest point, we can use the principle of conservation of energy. At the highest point, the potential energy was maximum, and at the lowest point, it's zero. Therefore, the total mechanical energy (sum of potential and kinetic energy) is conserved.
Total Mechanical Energy (E) = PE + KE
At the highest point: E = PE_max
At the lowest point: E = KE_max
Since the total mechanical energy is conserved, we have:
PE_max = KE_max
We've previously calculated that the maximum potential energy (PE_max) is approximately 1.412 Joules. Therefore:
KE_max = 1.412 Joules
So, the maximum kinetic energy of the pendulum is approximately 1.412 Joules.