A 0.200-m uniform bar has a mass of 0.750 kg and is released from rest in the vertical position, as the drawing indicates. The spring is initially unstrained and has a spring constant of k = 25.0 N/m. Find the tangential speed with which end A strikes the horizontal surface.
To find the tangential speed at which end A of the uniform bar strikes the horizontal surface, we need to use the principles of conservation of energy.
Here are the steps to find the answer:
Step 1: Determine the potential energy at the starting point (vertical position).
The potential energy at the starting point can be calculated using the formula:
PE = mgh
Where:
m = mass of the bar = 0.750 kg
g = acceleration due to gravity = 9.8 m/s^2 (approximate value)
h = height of the bar from the starting point to the horizontal surface
Since the bar starts from rest in the vertical position, the height h is equal to the length of the bar, which is given as 0.200 m.
PE = (0.750 kg) * (9.8 m/s^2) * (0.200 m)
PE = 1.470 J (joules)
Step 2: Determine the potential energy at the lowest point (when end A strikes the horizontal surface).
At the lowest point, all the potential energy is converted into kinetic energy. The kinetic energy can be calculated using the formula:
KE = (1/2)mv^2
Where:
m = mass of the bar = 0.750 kg
v = tangential speed of end A
Since the bar is horizontal at the lowest point, the height h is zero, and there is no potential energy remaining.
KE = 0 J (joules)
Step 3: Apply the conservation of energy principle.
According to the conservation of energy principle, the initial potential energy at the starting point is equal to the final kinetic energy at the lowest point. Therefore:
PE(starting point) = KE(lowest point)
1.470 J = (1/2)(0.750 kg)(v^2)
Step 4: Solve for v (tangential speed).
Rearrange the equation to solve for v:
v^2 = (2 * 1.470 J) / (0.750 kg)
v^2 = 3.92 J/kg
Take the square root of both sides to find v:
v = √(3.92 J/kg)
Using appropriate units for energy (joules) and mass (kilogram), the tangential speed v can be found.
Ensure that your calculator is set to use appropriate units and calculate the square root correctly.
v ≈ 1.98 m/s (approximate value)
Therefore, the tangential speed with which end A strikes the horizontal surface is approximately 1.98 m/s.