Two gerbils run in place with a linear speed of 0.60m/s on an exercise wheel that is shaped like a hoop.Find the angular momentum of the system if each gerbil has a mass of 0.22kg and the exercise wheel has a radius of 8.5 cm and a mass of 5.0 g.
L=I*W
w=v/r=0.6/0.085=7.06
I=M*r^2=0.005*(0.085^2)
L=2.55*10^-4
Well, well, well! Two gerbils on an exercise wheel, eh? Sounds like the start of a furry circus! Let's get down to business and calculate the angular momentum of this dynamic duo.
First, we need to calculate the total moment of inertia of the system. The moment of inertia can be calculated using the formula I = m*r^2, where I is the moment of inertia, m is the mass, and r is the radius.
For each gerbil, the moment of inertia would be:
I_gerbil = m_gerbil * r^2
= 0.22 kg * (0.085 m)^2
= 0.001429 kg⋅m^2
Next, for the exercise wheel itself, we need to add its moment of inertia:
I_wheel = m_wheel * r^2
= 0.005 kg * (0.085 m)^2
= 0.000036125 kg⋅m^2
Now, to find the total moment of inertia of the system, we simply add the moments of inertia together:
I_total = 2 * I_gerbil + I_wheel
= 2 * 0.001429 kg⋅m^2 + 0.000036125 kg⋅m^2
= 0.00289425 kg⋅m^2
Finally, we can calculate the angular momentum of the system using the formula L = I * ω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. In this case, the angular velocity can be calculated from the linear speed using the equation v = ω * r, where v is the linear speed and r is the radius of the hoop.
Ringing in the numbers, we have:
v = 0.60 m/s
r = 0.085 m
Solving for ω, we get:
ω = v / r
= 0.60 m/s / 0.085 m
≈ 7.06 rad/s
Now the final calculation:
L = I_total * ω
= 0.00289425 kg⋅m^2 * 7.06 rad/s
≈ 0.0204 kg⋅m^2/s
So, the angular momentum of this gerbil circus on wheels is approximately 0.0204 kg⋅m^2/s. Happy spinning, my furry friends!
To find the angular momentum of the system, we need to first calculate the individual angular momenta of the gerbils and the exercise wheel, and then sum them up.
The angular momentum (L) of an object can be calculated using the formula: L = Iω, where I is the moment of inertia and ω is the angular velocity.
Let's start by calculating the moment of inertia of each object.
For the exercise wheel:
The moment of inertia of a hoop (I) is given by the formula: I = mR², where m is the mass of the wheel and R is the radius.
Given:
Mass of the exercise wheel (m) = 5.0 g = 0.005 kg
Radius of the exercise wheel (R) = 8.5 cm = 0.085 m
Plugging these values into the formula, we can calculate the moment of inertia of the wheel:
I_wheel = m × R² = 0.005 kg × (0.085 m)² = 0.03425 kg·m²
Now let's calculate the moment of inertia of each gerbil assuming they are point-like objects (considering their mass concentrated at a single point):
For each gerbil:
Mass of the gerbil (m_gerbil) = 0.22 kg
Radius of the exercise wheel (R) = 0.085 m
Using the formula for the moment of inertia of a point-like object:
I_gerbil = m_gerbil × R² = 0.22 kg × (0.085 m)² = 0.0017735 kg·m²
Now we need to calculate the angular velocity (ω) of the system.
Given:
Linear velocity of the gerbils (v) = 0.60 m/s
Radius of the exercise wheel (R) = 0.085 m
The angular velocity can be calculated using the formula: v = Rω
So, ω = v/R = 0.60 m/s / 0.085 m = 7.0588 rad/s approx.
Now that we have the moment of inertia and angular velocity for each object, we can calculate the angular momentum (L) of each and then sum them up.
For the exercise wheel:
L_wheel = I_wheel × ω = 0.03425 kg·m² × 7.0588 rad/s ≈ 0.2418 kg·m²/s
For each gerbil (there are two gerbils in the system):
L_gerbil = I_gerbil × ω = 0.0017735 kg·m² × 7.0588 rad/s ≈ 0.0125 kg·m²/s
To find the total angular momentum of the system, we add up the individual angular momenta:
Total L = 2 × L_gerbil + L_wheel = 2 × 0.0125 kg·m²/s + 0.2418 kg·m²/s ≈ 0.2668 kg·m²/s
Therefore, the angular momentum of the system is approximately 0.2668 kg·m²/s.
To find the angular momentum of the system, we first need to find the moment of inertia of the exercise wheel and then calculate the angular velocity.
1. Moment of Inertia (I):
The moment of inertia of a hoop can be calculated using the formula:
I = M * R^2
Where:
I is the moment of inertia,
M is the mass of the object, and
R is the radius.
Given that the radius of the exercise wheel is 8.5 cm and the mass is 5.0 g, we need to convert the mass to kg:
Mass of the wheel = 5.0 g = 0.005 kg
Substituting these values into the formula:
I = 0.005 kg * (0.085 m)^2
I = 0.005 kg * 0.007225 m^2
I = 3.6125 x 10^-5 kg·m^2
2. Angular Velocity (ω):
The linear speed of the gerbils on the exercise wheel (v) is given as 0.60 m/s. The linear velocity of a point on the hoop is related to the angular velocity (ω) by the equation:
v = ω * R
Where:
v is the linear velocity,
ω is the angular velocity, and
R is the radius.
Rearranging the equation to solve for ω:
ω = v / R
ω = 0.60 m/s / 0.085 m
ω = 7.0588 rad/s
3. Angular Momentum (L):
Angular momentum (L) is given by the formula:
L = I * ω
Where:
L is the angular momentum,
I is the moment of inertia, and
ω is the angular velocity.
Substituting the values we calculated earlier:
L = 3.6125 x 10^-5 kg·m^2 * 7.0588 rad/s
L = 2.549 x 10^-4 kg·m^2·rad/s
Therefore, the angular momentum of the system is 2.549 x 10^-4 kg·m^2·rad/s.