Consider the figure above, where a rigid beam of negligible mass and 10 m long is supported by a cable attached to a spring. When NO block is hung from the beam, the length L (cable-spring) is equal to 5 m. Assume that immediately after block (weight of 280 N) is hung at the end of the beam the spring does not stretch, calculate the tension under this assumption. Using this tension find how much the spring stretches. Express the amount the spring is stretched in cm. k =8400 N/m.
h=4m
Spring at angle with unknown L, length, with block hanging and unknown angle
3m to spring end and 7m rest of horizontal rod with the block at the end of rod at 7m end.
To solve this problem, we need to analyze the forces acting on the beam and the spring.
First, let's consider the forces acting on the beam when no block is hung from it. Since the beam is in equilibrium, the tension in the cable is equal to the weight of the beam. We can set up an equation to represent this:
Tension = Weight of the Beam
Next, when a block of weight 280 N is hung from the end of the beam, the beam will experience an additional force due to the weight of the block. This will cause the cable to stretch and result in a new equilibrium position.
To find the tension in the cable under the assumption that the spring does not stretch, we need to consider the forces acting on the system. The tension in the cable will be equal to the sum of the weight of the beam and the block.
Tension = Weight of the Beam + Weight of the Block
Given that the weight of the block is 280 N, we need to find the weight of the beam. To do this, we can use the given dimensions of the beam. The horizontal distance from the point where the spring is attached to the block is 7 m, and the vertical distance (height) is 4 m. Since the beam is assumed to be of negligible mass, the weight of the beam can be calculated as the product of the length of the beam and the weight per unit length:
Weight of the Beam = Length of the Beam * Weight per Unit Length
Given that the length of the beam is 10 m and the weight per unit length is unknown, we need to determine the weight per unit length. This can be found by dividing the weight of the block by the horizontal distance:
Weight per Unit Length = Weight of the Block / Horizontal Distance
Now we can substitute the given values into the equations:
Weight per Unit Length = 280 N / 7 m = 40 N/m
Weight of the Beam = 10 m * 40 N/m = 400 N
Therefore, the tension in the cable under the assumption that the spring does not stretch is:
Tension = 400 N + 280 N = 680 N
To find how much the spring stretches, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The equation for this is:
F = -k * x
Where F is the force, k is the spring constant, and x is the displacement.
In this case, the force acting on the spring is the tension in the cable. We can solve for the displacement x:
-680 N = -8400 N/m * x
Solving for x:
x = 680 N / 8400 N/m = 0.081 m = 8.1 cm
Therefore, the spring stretches by 8.1 cm.