A ruler stands vertically against a wall. It is given a tiny impulse at θ=0∘ such that it starts falling down under the influence of gravity. You can consider that the initial angular velocity is very small so that ω(θ=0∘)=0. The ruler has mass m= 100 g and length l= 35 cm. Use g=10 m/s2 for the gravitational acceleration, and the ruler has a uniform mass distribution. Note that there is no friction whatsoever in this problem. (See figure)

(b) What is the force exerted by the wall on the ruler when it is at an angle θ=30∘? Express your answer as the x component Fx and the y component Fy (in Newton)

Fx=

Fy=

(c) At what angle θ0 will the falling ruler lose contact with the wall? (0≤θ0≤90∘; in degrees) [hint: the ruler loses contact with the wall when the force exerted by the wall on the ruler vanishes.]

θ0=

I=1/3*m*L^2

Eini= mg(L/2) + 0
Efin= mg(L/2)cos30 + 1/2*I*w^2
Eini=Efin ->
w=sqrt(3*g (1-cos(theta))/L)

b)
alpha=3*mg/2*sin(theta)/L
ax = -L/2*sin(theta) w^2 + L/2*cos(theta) alpha
Fx=m*ax
Fy=m*g*cos(theta)-m*w^2*(L/2)

c)
cos(theta)=2/3
theta=48.19

To solve this problem, we can start by analyzing the forces acting on the ruler at an angle θ. The forces acting on the ruler are the gravitational force and the force exerted by the wall.

(a) Let's first find the gravitational force acting on the ruler. The gravitational force is given by Fg = mg, where m is the mass of the ruler and g is the acceleration due to gravity.

Given:
m = 100 g = 0.1 kg
g = 10 m/s^2

Fg = (0.1 kg)(10 m/s^2)= 1 N

The gravitational force acting on the ruler is 1 N.

(b) Now let's find the components of the force exerted by the wall when the ruler is at an angle of 30°. The force exerted by the wall has both x and y components.

Fx: The x component of the force exerted by the wall is equal to the gravitational force because the ruler does not experience any net force in the horizontal direction. Therefore,

Fx = Fg = 1 N

Fy: The y component of the force exerted by the wall is equal to the component of the gravitational force perpendicular to the wall. At an angle of 30°, the gravitational force can be decomposed into its components as follows:

Fy = Fg * sin(θ)

Fy = (1 N)(sin(30°)) = 0.5 N

Therefore,
Fx = 1 N
Fy = 0.5 N

(c) To find the angle θ0 at which the ruler loses contact with the wall, we need to determine when the y component of the force exerted by the wall becomes zero.

Fy = 0.5 N

The y component of the force exerted by the wall becomes zero when sin(θ0) = 0.

Since sin(θ) = 0 when θ = 0° or θ = 180°, we can conclude that the ruler loses contact with the wall at an angle of θ0 = 0°.

Therefore,
θ0 = 0°