You want to make an old-fashioned pendulum clock, with a period of exactly 1.4 second(s). Use the Earth's gravitational acceleration g = 9.8 m/s2.

a) Ignoring the size of the pendulum bob, from what length wire should you hang your pendulum bob?
.4865 m

b) If you release the bob from an angle of 3.0°, what is the approximate acceleration of the bob at release? (you can use the approximation, sin(θ) = θ)
m/s2

c) Describing a pendulum as a harmonic oscillator requires the use of the small angle approximation. What is the magnitude of the difference between the actual acceleration (without using the small angle approximation) and the approximate acceleration (using the small angle approximation)? The answer is a very small number. As for all other numerical answers, provide your answer with 4 significant digits.
m/s2

d) From this release angle, what is the change in the bob's height from release to the bottom of its arc?
m

e) What is the bob's speed at the bottom of its arc? (use conservation of energy)
m/s

answered first not sure what formula to use on the rest can someone help?

b i got a=gsin(3)=.5129

a.

use
T=2πsqrt(L/g) to get
L=g(T/2π)^2
I get .4865 m also.

b.
At a release angle of 3°, the acceleration is g sin(θ).

c.
As given in the question, the difference is
Δa=(gθ - g sin(θ) )
where θ is measured in radians.
I get 0.0002 m/s²

d.
L(1-sin(θ)
I get 0.46.

e.
Equate energies:
mgL(1-sin(&theta))=(1/2)mv²
solve for v.

Parts d and e should read:

d.
L(1-cos(θ)
I get 0.000667 m.

e.
Equate energies:
mgL(1-cos(θ))=(1/2)mv²
solve for v.
v=sqrt(2*9.8*.4865*(1-cos(3°))
I get 0.11 m/s

Certainly! I'll guide you through the steps to find the solutions for the rest of the questions.

b) To find the approximate acceleration of the bob at release, we can use the formula for the acceleration of a pendulum at small angles:

a = g * θ

Since it is mentioned that we can use the approximation sin(θ) = θ, we can replace θ with 3.0° in radians (θ = 3.0° * π/180) and calculate the approximate acceleration.

c) To find the difference between the actual acceleration and the approximate acceleration, we can subtract the approximate acceleration (from part b) from the actual acceleration. The actual acceleration can be calculated using the formula:

a = -g * sin(θ)

where θ is the release angle in radians.

d) The change in the bob's height from the release to the bottom of its arc can be found using the conservation of mechanical energy. The total mechanical energy (E) is the sum of potential energy (PE) and kinetic energy (KE), given by:

E = PE + KE

Since the bob is released from rest at a certain height, the initial kinetic energy is zero. At the bottom of the arc, the potential energy is zero. Therefore, we can equate the initial potential energy to the final kinetic energy and solve for the change in height.

e) To find the bob's speed at the bottom of its arc, we can again use the conservation of mechanical energy. At the top of the arc, the bob has only potential energy, and at the bottom of the arc, it has only kinetic energy. We can equate the initial potential energy to the final kinetic energy and solve for the speed.

Let me know which specific question you would like help with, and I can provide further guidance and the respective formulas.