Find angle to the rods vertical

[dorian_stokes]

Homework Statement

A rod of mass 4.5 kg and length 1.2 m hangs from a hinge as shown in the figure below. The end of the rod is then given a "kick" so that it is moving at a speed of 5 m/s. How high will the rod swing? Express your answer in terms of the angle the rod makes with the vertical.

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi dorian_stokes!

Hint: use conservation of energy.

 

[kerimek]

To find the angle the rod makes with the vertical, we can use the conservation of energy principle. At the initial state, the rod has only gravitational potential energy, given by mgh, where m is the mass of the rod, g is the acceleration due to gravity, and h is the height. At the final state, the rod has both kinetic and potential energy. The kinetic energy is given by (1/2)mv^2, where v is the speed of the rod. The potential energy is given by mgh, where h is the height of the rod at the final state. Since energy is conserved, we can equate the initial and final energies and solve for h.

Initial energy = Final energy

mgh = (1/2)mv^2 + mgh

Canceling out the mass and solving for h, we get:

h = (1/2)v^2/g

Substituting the given values, we get:

h = (1/2)(5 m/s)^2/9.8 m/s^2

h = 1.275 m

To find the angle the rod makes with the vertical, we can use trigonometry. The height of the rod, h, is the opposite side of the triangle formed by the rod and the vertical. The length of the rod, 1.2 m, is the adjacent side of the triangle. Therefore, we can use the tangent function to find the angle, θ, between the rod and the vertical:

tanθ = h/1.2 m

θ = tan^-1(h/1.2 m)

Substituting the calculated value of h, we get:

θ = tan^-1(1.275 m/1.2 m)

θ = 49.3°

Therefore, the angle the rod makes with the vertical is approximately 49.3°.