Question

Solve the equations 1/2 mv² + 1/2 iω² = mgh and v = rω for the speed v using substitution, given that i = mr² and h = 3.57 m. (note that mass m and radius r will both cancel, so their numerical values aren't required.)

Answer 1

To solve the given equations using substitution, we can start by substituting the values of i and h. Simplify the resulting equation by combining like terms and dividing by (3/2)m. Finally, take the square root of both sides to find the speed v in terms of g and h.

Step-by-step explanation:

We can solve the equations using substitution. Let's start by substituting the value of i with mr² and h with 3.57m.

Substituting these values, we get:

(1/2)mv² + (1/2)(mr²)ω² = mg(3.57)

Also, we have the equation v = rω, which we can substitute into the above equation:

(1/2)mv² + (1/2)(mr²)(v/r)² = mg(3.57)

Simplifying this equation, we get:

(1/2)mv² + (1/2)m(v²/r²) = mg(3.57)

Combining like terms, we have:

(3/2)mv² = mg(3.57)

Dividing both sides by (3/2)m, we get:

v² = g(3.57)

Taking the square root of both sides, we find:

v = √(g(3.57))

So the speed v is equal to the square root of g times 3.57.