the speed of Simple Pendulum ?

Question

The figure shows a simple pendulum of length L = 41 cm and mass m = 2.1 kg. It's bob is observed to have a speed of vo = 3.2 m/s when the cord makes an angle θo = 30°.
a.) What is the speed of the bob when it is in its lowest position?
b.) What is the least value that v0 must have if the cord is to swing up to a horizontal position?

Answer 1

Use energy methods. Consider a Cartesian coordinate system where the pivot of the pendulum is at the origin. In particular, notice that the reference height for gravitational potential energy is at the pivot, which is why the gravitational potential energy terms below are negative. You could choose a reference at the bottom of the pendulum's arc, but that would make the problem a bit messier.



solve for v:

1/2 m v0^2 - mgL cos(θ0) = 1/2 m v^2 - mgL



Solve for v0:

1/2 m v0^2 - mgL cos(θ0) = 0

Answer 2

Your instructor is right that it somewhat is easiest to unravel this situation in terms of ability. yet your physique of recommendations is okay besides, see you later as you employ it wisely. i think you're commencing from d = A*cos(2pi*f*t), the displacement d as a function of time. in case you start up off there, then the time spinoff is v, the speed. if so displacement must be the arc-length displacement from the middle place, so it somewhat is L * theta, the place theta = angular displacement in radians. the optimal amplitude A is as a result a million.0 * (pi/6) = pi/6 meters. the optimal speed is A * w the place w = angular frequency. For an common pendulum the era is two*pi*sqrt(L/g), so the frequency = a million/T = sqrt(g/L) / (2*pi) and the angular frequency w = 2*pi*f = sqrt(g/L) = sqrt(9.8 m/sec^2/ a million.0 m) = 3.13 rad/sec. as a result the optimal speed is a million.0 * pi/6 * 3.13 = a million.sixty 4 m/sec. Oh, and in answer to your direct question, if a pendulum of length L is displaced by using an perspective theta, the tip of the pendulum is a vertical distance of L*cos(theta) under the pivot. only draw the triangle with the pendulum because of fact the hypotenuse. because of fact it exchange into initially a distance L under the pivot, this is been raised by using a top L - L*cos(theta) = L(a million - cos(theta)). And a million - cos(30) = 0.134.

Answer 3

a) use conservation of energy: let h = L(1 - cos 30) then the energy at 30 degrees is the kinetic energy 1/2 m v0^2 plus the potential energy mgh. At the lowest position the total energy is the same but it is all kinetic: E = 1/2 mv^2 = 1/2mv0^2 + mgh You know everything on the right side so you can solve for the velocity v.