A ladder leaning against a wall is falling. The ladder is of length, L, and at this instant the ladder is at an angle, theta(Ø), with respect to the wall, and the horizontal velocity of the base of the ladder ,v_A, is known. All answers are I'm the terms given , fixed coordinate system, OXYZ.( Point A is the ladder bottom, Point C is the center, and point B is the top of the ladder.(the point in contact with the wall)
1)What is the velocity, at this instant , of point C in terms of v_A, L, theta(Ø), and g?
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2)What is the velocity, at this instant , of point B in terms of v_A, L, theta(Ø), and g?
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To find the velocities at point C and point B in terms of the given variables, we will use the concepts of rotational motion and kinematics.
1) Velocity of Point C:
To find the velocity of point C, we need to consider the rotational motion of the ladder about point B. We can use the relationship between linear and angular velocities.
The linear velocity of point C can be calculated as the product of the angular velocity and the distance from the pivot point B to point C. Since the velocity at the base of the ladder is given as v_A, which is a linear velocity, we need to find the angular velocity first.
The angular velocity of the ladder can be obtained from its angular displacement with respect to time. Assuming the ladder is falling vertically, the angular displacement can be given as theta(Ø). Therefore, the angular velocity is given by:
ω = (θ / t)
Next, we can determine the distance from the pivot point B to point C. Since the ladder length is given as L, using trigonometry, the distance is:
BC = L * cos(theta(Ø))
Then, the velocity of point C can be calculated using the formula:
vC = ω * BC
Substituting the values obtained above, we have:
vC = ((θ / t) * L * cos(theta(Ø)))
2) Velocity of Point B:
To find the velocity of point B, we can consider the linear velocity of point C and the fact that point B is fixed on the wall.
The velocity of point B is the horizontal component of the velocity of point C. Since the ladder is falling vertically, the horizontal component of vC is equal to the horizontal velocity at point A (v_A).
Hence, the velocity of point B can be expressed as:
vB = v_A
In summary, the velocities at point C and point B in terms of the given variables (v_A, L, theta(Ø), and g) are:
1) Velocity of point C: vC = ((θ / t) * L * cos(theta(Ø)))
2) Velocity of point B: vB = v_A