A 0.34 kg pendulum bob is attached to a string 1.2 m long. What is the change in the gravitational potential energy of the system as the bob swings from the bottom of the string, where θ = 37°?
i don't know how to do this. The bottom of the string is 1.2 m and it moves 37°...
Use trigonometry to figure out the displacement in the y direction
(1.2 m) * (cos 37) = displacement in the y direction
Displacement in x direction is not a concern here since we are talking about the gravitational force
the gives you delta y
then use you formula for potential energy due to y displacement to get change on gravitaitonal potential energy I think it is force* delta y * mass
mgy = gravitational potential energy
The force in this case is 9.8 (that due to gravity)
To calculate the change in gravitational potential energy, we need to know the initial and final heights of the pendulum bob.
In this case, the initial height is the lowest point of the bob's swing, which we can take as the bottom of the string, where the θ (angle) is given as 37°. The final height is the highest point of the swing, where the bob reaches its maximum height.
To find the initial height, we can use the trigonometric relationship between the length of the string and the angle made with the vertical line. Given that the length of the string is 1.2 m and the angle is 37°, we can calculate the initial height (h1) using the sine function:
sin θ = h1 / L
where:
θ = 37°
L = 1.2 m (length of the string)
Rearranging the equation to solve for h1, we have:
h1 = L × sin θ
Substituting the values:
h1 = 1.2 m × sin 37°
Calculating this, we find:
h1 ≈ 1.2 m × 0.6018 ≈ 0.72216 m
Now, to find the final height (h2), we need to determine the maximum height reached by the bob. At the highest point, the pendulum momentarily stops before swinging back downwards. Hence, the velocity of the bob at that point is zero.
Using the principle of conservation of energy, we know that the total mechanical energy of the system is conserved. Therefore, at the highest point, the sum of the gravitational potential energy and the kinetic energy is equal to the sum at the lowest point.
Since the velocity is zero at the highest point, the kinetic energy is also zero. Therefore, the total mechanical energy at the highest point is equal to the gravitational potential energy.
Now, the gravitational potential energy (PE) is given by:
PE = mgh
where:
m = 0.34 kg (mass of the bob)
g = 9.8 m/s² (acceleration due to gravity)
h = h2 (final height)
Substituting the values, the gravitational potential energy at the highest point is:
PE = (0.34 kg) × (9.8 m/s²) × h2
At this point, we can equate the gravitational potential energies at the highest and lowest points to find the change in gravitational potential energy, ΔPE:
ΔPE = PE at highest point - PE at lowest point
= (0.34 kg) × (9.8 m/s²) × h2 - (0.34 kg) × (9.8 m/s²) × h1
Substituting the values we know:
ΔPE = (0.34 kg) × (9.8 m/s²) × h2 - (0.34 kg) × (9.8 m/s²) × 0.72216 m
Now, you can calculate the change in gravitational potential energy by substituting the value for h2. The value of h2 will depend on the amplitude of the pendulum's swing or the maximum angle it reaches, which you haven't provided.
Once you have the value for h2, substitute it into the equation and perform the calculation to find the change in gravitational potential energy.