A rod of mass 4.3 kg and length 4.3 m hangs from a hinge as shown in Figure P9.28. The end of the rod is then given a “kick” so that it is moving at a speed of 2.3 m/s. How high will the rod swing? Express your answer in terms of the angle (in degree) that the rod makes with the vertical.

Question

A rod of mass 4.3 kg and length 4.3 m hangs from a hinge as shown in Figure P9.28. The end of the rod is then given a “kick” so that it is moving at a speed of 2.3 m/s. How high will the rod swing? Express your answer in terms of the angle (in degree) that the rod makes with the vertical.

Hinge --> 0
i
i <-- Steel rod
i
-------> swings this way

Answer 1

w=2.3/4.3= 1 rad per second.

Now, the moment of Inertia for the rod swinging about an end is... 1/3 ml^2

initial KE= 1/2 I w^2=1/6 ml^2 figure all that out given m, l

now, final PE= mgh were h is to the center of mass. Drawing the figure, then

h+lcosTheta=lor
h= l(1-cosTheta)
finally we have
1/6ml^2=mgl(1-cosTheta)
L/6g=1-cosTheta
costheta= 1- 4.3/6g
theta= arc cos ( above)

check all this.

Answer 2

To solve this problem, we need to analyze the conservation of energy. The initial kinetic energy of the rod is converted to gravitational potential energy as it swings up.

First, let's calculate the initial kinetic energy of the rod. The formula for kinetic energy is:

KE = (1/2) * mass * velocity^2

Given that the mass of the rod is 4.3 kg and the velocity is 2.3 m/s, we can calculate the initial kinetic energy:

KE_initial = (1/2) * 4.3 kg * (2.3 m/s)^2

Next, let's determine the final potential energy of the rod at the highest point of its swing. The formula for gravitational potential energy is:

PE = mass * gravity * height

Since the rod stops momentarily at the highest point, all of its initial kinetic energy is converted to gravitational potential energy. Therefore, the initial kinetic energy is equal to the final potential energy:

KE_initial = PE_final

Substituting the formulas and values:

(1/2) * 4.3 kg * (2.3 m/s)^2 = 4.3 kg * gravity * height

Now we can solve for height:

height = (1/2) * (2.3 m/s)^2 / gravity

To express the result in terms of the angle (in degrees) that the rod makes with the vertical, we can use trigonometry. The height of the rod is equal to the length multiplied by the sine of the angle (θ):

height = length * sin(θ)

Substituting the known values, we have:

length * sin(θ) = (1/2) * (2.3 m/s)^2 / gravity

Simplifying the equation, we can isolate sin(θ):

sin(θ) = (1/2) * (2.3 m/s)^2 / (gravity * length)

Finally, we can find the angle (in degrees) using the inverse sine function:

θ = sin^(-1)[(1/2) * (2.3 m/s)^2 / (gravity * length)]

Calculating this expression will give us the angle in degrees that the rod makes with the vertical, which is equivalent to the height of the rod swing.