Suppose that a simple pendulum consists of a small 66.0 g bob at the end of a cord of negligible mass. Suppose that the angle between the cord and the vertical is given by
θ = (0.0800 rad) cos[(4.30 rad/s)t + ϕ]
(a) What is the pendulum's length?
(b) What is its maximum kinetic energy?
I figured out a) by using w=2pi/T
T=1.46 then used T to solved T=(2pi*sq root L/g) L = .53 but I can't figure out an equation to give me the velocity. Any ideas?
(a) w = 4.40 rad/s
is the angular frequency, sqrt(g/L)
Solve for L, the pendulum length
L = g/w^2
(b) Max velocity Vmax = = L*theta*w
Maximum KE = (1/2) M Vmax^2
You are right about L = 0.53. Don't forget the units (meters)
In simple harmonic motion,
max velocity = w*(Amplitude) and
max acceleration = w^2*(Ampitude)
In your case I had to add an L factor to get linear velocities from the angular amplitude.
Where did the Vmax = L*theta*w
come from? How do I find theta?
Theta should be theta-max, the angular amplitude, which is 0.0800 radians
Sorry about that
Vmax= (.53m)(.0800rad)(4.30 rad/s)
=.182
KE=1/2mv^2
=1/2(.066kg)(.182)^2
=.0011 or 1.09e-03
That doesn't look right to me. what am I doing wrong? ;_;
Another question, sorry, trying my bests to understand your thinking. I don't see where you knew to multiple L * theta * W.
I understand that the 2nd derivative of the equation gives you the Velocity function.
No, the first derivative of your theta vs t (multiplied by L) gives you the velocity function.
L*theta_max is the displacement amplitude (in small angle approximation) ; so
w*L*theta_max
is the maximum velocity
To find the answers to these questions, we need to use the given information about the simple pendulum and apply the relevant formulas. Let's break down the problem step by step.
(a) To find the pendulum's length, we can use the equation of motion for a simple harmonic oscillator:
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. However, in this case, the equation for the angle θ is given as a function of time, rather than the period. So, we need to convert the given information into the period.
The period of the pendulum is the time it takes for the pendulum to complete one full swing back and forth. It can be determined by the time taken for the angle θ to complete one full cycle. Therefore, we can consider the expression inside the cosine function as the argument for one full cycle:
(4.30 rad/s)t + ϕ = 2π
Rearranging the equation, we have:
t = (2π - ϕ)/(4.30 rad/s)
Now we have the period of the pendulum, which is the time taken for one complete cycle. Substituting this value into the equation for the period:
T = (2π√(l/g))
we can rewrite it as:
T = (2π√(l/9.8 m/s²))
Equating this with the period we found earlier:
(2π√(l/9.8 m/s²)) = (2π - ϕ)/(4.30 rad/s)
Simplifying the equation, we can solve for l:
l = (9.8 m/s²) * ((2π - ϕ)/(4.30 rad/s))²
Substitute the given value of ϕ into the equation and calculate l.
(b) To find the maximum kinetic energy of the pendulum, we can use the equation:
K.E. = (1/2)mv²
where m is the mass of the bob and v is its velocity. Since the angle θ is given as a function of time, we can find the velocity of the bob by taking the first derivative of the angle θ with respect to time:
v = dθ/dt
Differentiating the given expression for θ:
v = (-0.0800 rad * 4.30 rad/s * sin[(4.30 rad/s)t + ϕ])
Now we can substitute this expression for v into the equation for kinetic energy:
K.E. = (1/2) * (0.0660 kg) * [(-0.0800 rad * 4.30 rad/s * sin[(4.30 rad/s)t + ϕ])]²
Substitute the given values and calculate the maximum kinetic energy.
Remember to include appropriate units in your final answers.