What hanging mass will stretch a 1.9-m-long, 0.54 - diameter steel wire by 1.2 ?
Do you use units in your class?
SI units
Kendra, you did not put units in your question. There is no way to answer it. what is .54?
.54mm & 1.2mm
sorry
What hanging mass will stretch a 2.3-m-long, 0.70 - diameter steel wire by 1.5 ?
To determine the hanging mass that will stretch a steel wire by a certain amount, we need to use Hooke's law, which relates the force applied to a spring or wire to the displacement caused by that force.
Hooke's law states that the force (F) exerted by a spring is directly proportional to the displacement (x) caused by that force. It can be mathematically expressed as:
F = k * x
Where:
- F is the force applied to the spring or wire (in Newtons),
- k is the spring constant (in N/m), which represents the stiffness of the spring,
- x is the displacement caused by the force (in meters).
In this case, the elongation or displacement caused by the hanging mass is given as x = 1.2 m. To determine the mass required to cause this displacement, we first need to find the spring constant or stiffness of the wire.
The spring constant of a wire can be calculated using the formula:
k = (4 * L * E * r^2) / (π * D^4)
Where:
- L is the length of the wire (in meters),
- E is the modulus of elasticity of the material (in Pascals),
- r is the radius of the wire (in meters),
- D is the diameter of the wire (in meters).
In this case, we are given that the wire length (L) is 1.9 m, and the wire diameter (D) is 0.54 m. However, we need the radius (r) instead of the diameter, so we divide the diameter by 2:
r = D / 2 = 0.54 / 2 = 0.27 m
Now, we need to know the modulus of elasticity (E) of the steel wire. The modulus of elasticity represents how much a material stretches under a given amount of force. For steel, the approximate value of the modulus of elasticity is 200 × 10^9 Pa (Pascals).
Now, we can calculate the spring constant (k) using the formula:
k = (4 * L * E * r^2) / (π * D^4)
= (4 * 1.9 * 200 * 10^9 * (0.27^2)) / (π * (0.54^4))
Evaluating this expression, we find:
k ≈ 1.83 × 10^8 N/m
Finally, we can calculate the force (F) exerted by the hanging mass using Hooke's law:
F = k * x
= (1.83 × 10^8 N/m) * 1.2 m
Calculating this expression, we find:
F ≈ 2.20 × 10^8 N
To find the mass (m) that will create this force, we can use the formula:
F = m * g
Where:
- F is the force exerted by the hanging mass (in Newtons),
- m is the mass of the hanging mass (in kilograms),
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Rearranging the formula to solve for mass (m), we get:
m = F / g
= (2.20 × 10^8 N) / 9.8 m/s^2
Calculating this expression, we find:
m ≈ 2.24 × 10^7 kg
Therefore, a hanging mass of approximately 22,400,000 kg will stretch the steel wire by 1.2 m.