A child's top is held in place, upright on a frictionless surface. The axle has a radius of r= 2.21 mm. Two strings are wrapped around the axle, and the top is set spinning by applying T= 2.40 N of constant tension to each string. If it takes 0.830 s for the string to unwind.
1- How much angular momentum does the top acquire? Assume that the strings do not slip as the tension is applied.
2- If the final tangential speed of point P, h= 35.0 mm above the ground, is 1.15 m/s and the angle theta is 26.0 what is the top's moment of inertia?
To solve these problems, we will use the formulas related to angular motion and angular momentum.
1- Angular momentum (L) is defined as the product of the moment of inertia (I) and the angular velocity (ω). The angular momentum acquired by the top can be calculated using the formula:
L = I * ω
Where:
- L is the angular momentum
- I is the moment of inertia
- ω is the angular velocity (which is equal to the change in angle divided by the change in time)
In this case, the angular velocity can be calculated as:
ω = Δθ / Δt
Where:
- Δθ is the change in angle (assuming the top starts from rest, Δθ will be equal to the final angle)
- Δt is the change in time (the time it takes for the string to unwind)
Given that Δθ = 2π radians (360 degrees) and Δt = 0.830 s, we can calculate ω:
ω = 2π / 0.830
Now, we can calculate the angular momentum using the given moment of inertia (I) and the calculated ω.
2- The tangential speed at point P can be calculated using the formula:
v = r * ω
Where:
- v is the tangential speed at point P
- r is the radius of the axle
- ω is the angular velocity (which we already calculated in the previous step)
The moment of inertia (I) can be calculated using the formula:
I = m * r^2
Where:
- m is the mass of the top
- r is the radius of the axle
Given that the tangential speed v = 1.15 m/s, the radius r = 35.0 mm, and the angle θ = 26.0 degrees, we can calculate the moment of inertia I.
Now, let's go step-by-step to solve these problems:
1- Calculate Angular Momentum:
- Calculate ω:
ω = 2π / 0.830
ω ≈ 7.583 rad/s
- Calculate angular momentum (L):
L = I * ω
2- Calculate Moment of Inertia:
- Convert radius r from mm to m (r = 35.0 mm):
r = 35.0 mm / 1000
r = 0.035 m
- Calculate moment of inertia (I):
I = m * r^2
Now, you can plug in the values for the variables and calculate the respective values.
To answer the given questions, we need to understand the concepts of angular momentum and moment of inertia.
1. Angular Momentum:
Angular momentum (L) is a measure of the rotation of an object. It is defined as the product of the moment of inertia (I) and the angular velocity (ω) of the object.
L = Iω
In this case, the top is set spinning by applying a constant tension (T) to each string. The tension in the string exerts a torque (τ) on the top, causing it to rotate. Since the top is initially at rest, we can assume the initial angular velocity (ω) is zero.
To find the angular momentum acquired by the top, we first need to calculate the moment of inertia of the top.
2. Moment of Inertia:
The moment of inertia (I) represents an object's resistance to changes in its rotational motion. For a rigid body rotating about a fixed axis, the moment of inertia depends on the mass distribution and the axis of rotation.
In this case, the top is held upright on a frictionless surface, so it will rotate about its axle. The moment of inertia of the top can be calculated using the parallel-axis theorem:
I = I_cm + m*d^2
where I_cm is the moment of inertia about the center of mass, m is the mass of the top, and d is the perpendicular distance between the center of mass and the axis of rotation.
Now, let's solve each question separately:
1. To find the angular momentum acquired by the top, we need to calculate the moment of inertia (I) and the final angular velocity (ω).
Given:
- Radius of the axle (r) = 2.21 mm = 0.00221 m
- Tension applied to each string (T) = 2.40 N
- Time taken for the string to unwind (t) = 0.830 s
First, we need to find the final angular velocity (ω) using the formula:
ω = Δθ/Δt
Since the string unwinds completely, the angle Δθ is 2π radians (one complete revolution). Therefore,
ω = 2π/0.830
Now, we can calculate the moment of inertia (I) using the formula:
I = T/(α*r)
Where α is the angular acceleration, given by α = ω/t.
Substituting the values, we have:
α = ω/t = (2π/0.830) / 0.830
I = T/(α*r) = 2.40 / [(2π/0.830) / 0.830 * 0.00221]
2. To find the top's moment of inertia, we need to know the final tangential speed of point P, its height from the ground (h), and the angle theta.
Given:
- Final tangential speed of point P (v) = 1.15 m/s
- Height of point P from the ground (h) = 35.0 mm = 0.035 m
- Angle theta (θ) = 26.0 degrees
The tangential speed is related to the angular velocity (ω) and the radius (r) by the formula:
v = ω*r
Using the above formula, we can calculate the angular velocity (ω). Then, we can find the moment of inertia (I) using the following formula for a point mass rotating about an axis:
I = m*h^2/(sin^2(θ))
Where m is the mass of the top.
Now, with all the necessary information, you can calculate the angular momentum and moment of inertia for the given child's top.