Question

Solve the equations 1 2 mv2 1 2 i2 = mgh and v = r for the speed v using substitution, given that i = mr2 and h = 2.79 m.

Answer 1

To find the final speed v, we simplify the given equations by canceling out m and r, resulting in the equation v^2 = 2gh. Substituting in the values for g (acceleration due to gravity) and the given h (2.79 m), then taking the square root, we calculate the final speed v.

Step-by-step explanation:

To solve the equations 1/2 mv2 + 1/2 i2 = mgh and v = rω for the speed v using substitution, given the known moment of inertia i = mr2 and height h = 2.79 meters, we follow these steps:

1. First, substitute i = mr2 into the first equation, resulting in 1/2 mv2 + 1/2 mr2ω2 = mgh.
2. Rearrange this equation to express v solely in terms of m, g, h, and ω.
3. Since we also know that v = rω, we can substitute this relationship into the rearranged equation.
4. However, the cylinder's radius r and mass m cancel out in our given information, leading to the simplified equation 2gh = 2v2, which can be further simplified to gh = v2.
5. Now, substituting the known value for h, we can solve for v2 and find v by taking the square root.

The final step gives us the equation v2 = 2gh and by substituting h = 2.79 m and g = 9.8 m/s2 (standard acceleration due to gravity), we get v2 = 2 * 9.8 m/s2 * 2.79 m.

After calculating this, we take the square root to find the final speed v.