Prove torque equation 2sin^2(theta/2) * cos(theta + beta) = sin (beta)

In summary, the conversation discusses a problem involving a uniform smooth plank and a sphere of weight 2W placed on a smooth inclined plane. The goal is to prove that in the position of equilibrium, the angle between the plank and the plane can be represented by the equation 2sin^2(theta/2) * cos(theta + beta) = sin(beta). The conversation includes attempts at solving the problem, but the angle remains elusive.
  • #1
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Homework Statement



A uniform smooth plank weight W and length 2a is hinged to the bottom horizontal edge of a smooth, fixed plane inclined at angle (beta) to the horizontal. A sphere of radius (1/2)a and weight 2W is placed between the plank and the plane. Assume no friction. Prove that, in the position of equilibrium, the angle (theta) between the plank and the plane will be given by the equation

2sin^2(theta/2) * cos(theta + beta) = sin (beta)


Homework Equations



sum of torque =0
sum of force along(x and y) = 0

The Attempt at a Solution


I try to draw a FBD, but i don't know where to start to get to that equation :x.

(the pic is drawn not to scale)
 

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  • #2
this is what i try so far T.T *writing equation with blood*

sum of force along the x-axis of the ball = N(plank)(sin[theta]) - 2w(sin[beta]) = 0

N(plank) = 2w(sin[beta]) / (sin[theta])

take torque of the plank ( draw another separate diagram of the plank{with blood :P} ) axis on the tip of the plank(the bottom tip) *clockwise*

* a = half the length of the plank (assuming the plank is uniform)
*(a/2)(sin[theta/2]) = draw a triangle from the center of the ball to the tip of the plank, we then obtain sin(theta/2) = x/(a/2)
*sin[theta] come from the resultant of N(plank)y

= W(cos[theta + beta])(a) - N(plank)(a/2)(sin[theta])(sin[theta/2]) = 0

subtitude N(plank) and switch the n(plank) thingy to the other side

W(cos[theta + beta])(a) = 2w(sin[beta]) / (sin[theta]) * (a/2)(sin[theta/2])

simplify

cos(theta + beta) = sin(beta)*sin(theta/2)

...*poof! a miracle happen*

2sin^2(theta/2)*cos{theta + beta) = sin(beta)

T.T somebody help me...i think i mess up on the angle D:
 
  • #3
:X no one helping me >.<

i really having a hard time figuring out the angle. T.T
 

1. What is the torque equation?

The torque equation is a mathematical formula that relates the rotational force, or torque, applied to an object to its moment of inertia and angular acceleration. It is commonly used in physics and engineering to calculate the amount of force needed to cause an object to rotate.

2. How do you prove the torque equation?

To prove the torque equation, you would need to use principles of trigonometry and calculus to manipulate the equation and show that it follows from the fundamental laws of rotational motion, such as Newton's second law and the definition of torque as the cross product of force and distance. The specific steps to prove the equation may vary depending on the context and assumptions made.

3. What does the variable theta represent in the torque equation?

The variable theta in the torque equation represents the angle between the force vector and the lever arm, which is the distance from the axis of rotation to the point where the force is applied. This angle is important in determining the direction and magnitude of the torque on an object.

4. What is the significance of the variable beta in the torque equation?

The variable beta in the torque equation represents the angle between the lever arm and the direction of the force. This angle is important in determining the direction of the torque on an object. In some cases, beta may be equal to theta, but in others, it may differ and affect the resulting torque.

5. Can the torque equation be applied to all types of rotational motion?

Yes, the torque equation can be applied to all types of rotational motion, including uniform circular motion, rotational acceleration, and rotational equilibrium. However, the specific form of the equation may vary depending on the situation and the assumptions made. For example, in the case of rotational equilibrium, the torque equation would be set equal to zero, as there is no net torque acting on the object.

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