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 = 4.20 m.
(Note that mass m and radius r will both cancel, so their numerical values aren't required.)
To solve the equations and find the value of v, let's substitute the given values and known relationships between the variables.
We are given the equations:
1) 1/2 * mv^2 + 1/2 * I * ω^2 = mgh
2) v = rω
We are also given the following relationships:
I = mr^2
h = 4.20 m
We can start by substituting the value of I in equation 1:
1/2 * mv^2 + 1/2 * (mr^2) * ω^2 = mgh
Now, let's focus on the term 1/2 * mv^2. Since we have the equation v = rω, we can substitute the value of v with rω:
1/2 * m(rω)^2 + 1/2 * (mr^2) * ω^2 = mgh
Simplifying this equation further:
1/2 * m(r^2ω^2) + 1/2 * (mr^2) * ω^2 = mgh
Combining like terms:
1/2 * m(r^2ω^2 + r^2ω^2) = mgh
Simplifying this equation even more:
m(r^2ω^2) = mgh
Now, we can cancel out the mass, m, from both sides of the equation:
r^2ω^2 = gh
Finally, let's substitute the given value of h = 4.20 m into the equation:
r^2ω^2 = g * 4.20
Since we're looking for the value of v, we can rearrange equation 2 (v = rω) to express ω in terms of v:
ω = v/r
Substituting this value into our previous equation:
r^2(v/r)^2 = g * 4.20
Simplifying further:
v^2 = gr^2 * 4.20
Taking the square root of both sides to solve for v:
v = √(gr^2 * 4.20)
Therefore, the speed v can be found by taking the square root of gr^2 multiplied by 4.20.