a ladder AP of length 5m inclined to a vertical wall is slipping over a horizontal surface with velocity 2m/s , when A is at a distance 3m from ground . what is the velocity of centre of mass at this moment?
To find the velocity of the center of mass of the ladder at this moment, we can use the concept of relative motion.
First, let's break the velocity of the ladder into two components:
1. Vertical component: As ladder is inclined to the vertical wall, the vertical component of the velocity does not change with time. Given that the ladder is slipping over a horizontal surface with a velocity of 2 m/s and A is at a distance of 3 m from the ground, the vertical component of velocity can be calculated as:
Vertical velocity = velocity * sin(θ)
where θ = angle of inclination of the ladder with the horizontal
Since the ladder is inclined to the vertical wall and is slipping over a horizontal surface, the angle between the ladder and the horizontal surface is the same as the angle of inclination with the vertical wall.
Therefore, we can calculate the vertical velocity component as:
Vertical velocity = 2 m/s * sin(θ)
2. Horizontal component: The horizontal component of velocity is due to the slipping motion and changes with time. Given that the horizontal component of velocity is 2 m/s, it remains constant throughout.
Now, let's find the angle of inclination (θ) and the vertical velocity component.
Consider a right-angled triangle with the ladder as the hypotenuse and the horizontal distance from the wall as the base. The vertical distance from the ground to point A can be considered as the height of the triangle.
Using Pythagoras' theorem, we can find the length of the ladder:
Length of the ladder = √[(distance from the wall)^2 + (height)^2]
Given that the length of the ladder is 5 m and the distance from the wall is 3 m, we can solve for the height:
Height = √[(Length of the ladder)^2 - (distance from the wall)^2]
Once we have the height and the distance from the wall, we can calculate the angle of inclination using trigonometric ratios:
sin(θ) = height / length of the ladder
Finally, with the value of sin(θ), we can calculate the vertical velocity component:
Vertical velocity = 2 m/s * sin(θ)
The center of mass of the ladder moves vertically at the same velocity as point A, and horizontally with the same component as point A due to slipping. Therefore, the velocity of the center of mass at this moment will be equal to the vertical velocity calculated as well as the horizontal component of the velocity (which is 2 m/s).
To find the velocity of the center of mass of the ladder at the given moment, we can use the principle of conservation of linear momentum. Since there are no external forces acting on the system, the initial momentum of the system when the ladder is slipping should be equal to the momentum at the given moment.
Let's break down the problem step-by-step:
Step 1: Find the initial momentum of the system:
The initial momentum, P_initial, can be calculated as the product of the mass and velocity.
P_initial = m * v_initial
Step 2: Find the final momentum of the system:
The final momentum, P_final, can also be calculated as the product of the mass and velocity. However, in this case, the velocity will be different since the ladder is sliding and changing its position.
P_final = m * v_final
Step 3: Set the initial momentum equal to the final momentum:
P_initial = P_final
m * v_initial = m * v_final
Step 4: Solve for the final velocity of the center of mass, v_final:
Since the mass (m) is common on both sides of the equation, it cancels out.
v_initial = v_final
Therefore, the velocity of the center of mass at the given moment when A is at a distance of 3m from the ground is 2m/s.
Draw the diagram.
label the base of the ladder A, the top P, the corner O, and the cg C.
OA=3, AP=5, and you can compute OP as 4
OA^2+OP^2=AB^2
2 OA doa/dt +2 OP dop/dt=0 by taking the derivative (AB is a constant)
well, you know OA, OP, doa/dt=2, you can find dop/dt. At that instant, the center C has a velocity (1/2 doa/dt horizontal, and 1/2 dop/dt vertical)
so you have to add those two components as a vector