A toy top consists of a rod with a diameter of 13.00 mm and a disk of mass 0.0160 kg and a diameter of 3.5 cm. The moment of inertia of the rod can be neglected. The top is spun by wrapping a string around the rod and pulling it with a velocity that increases from zero to 3.0 m/s over 0.40 s.

a) what is the resulting angular velocity in rad/s?

b) what is the force exerted on the string in newtons?

0.00125

To find the resulting angular velocity, we can use the equation:

Angular velocity (ω) = Change in angle / Change in time

In this case, we know that the velocity of the string increases from zero to 3.0 m/s over 0.40 s. Therefore, we can calculate the change in angle using the formula:

Change in angle = (Final velocity - Initial velocity) / (Change in time)

a) Calculating the resulting angular velocity (ω):

Change in angle = (3.0 m/s - 0 m/s) / 0.40 s
Change in angle = 7.5 rad/s

So, the resulting angular velocity is 7.5 rad/s.

To find the force exerted on the string, we can use the formula:

Force (F) = Mass (m) x Acceleration (a)

We can find the acceleration using the formula:

Acceleration = Change in velocity / Change in time

b) Calculating the force exerted on the string (F):

Acceleration = (Final velocity - Initial velocity) / (Change in time)
Acceleration = (3.0 m/s - 0 m/s) / 0.40 s
Acceleration = 7.5 m/s^2

Now, we can calculate the force exerted on the string:

Force (F) = Mass (m) x Acceleration (a)
Force (F) = 0.0160 kg x 7.5 m/s^2
Force (F) = 0.12 N

So, the force exerted on the string is 0.12 N.

To find the resulting angular velocity of the toy top, we can use the principle of conservation of angular momentum.

Angular momentum (L) is given by the formula L = Iω, where I is the moment of inertia and ω is the angular velocity.

In this case, the moment of inertia of the toy top consists of two components: the disk and the rod. Since the moment of inertia of the rod is neglected, we only need to consider the moment of inertia of the disk.

The moment of inertia of a uniform disk can be calculated using the formula I = (1/2) * m * r^2, where m is the mass of the disk and r is the radius.

First, let's convert the diameter of the disk and rod into their respective radii:
Radius of the disk (r) = diameter / 2 = 3.5 cm / 2 = 1.75 cm = 0.0175 m
Radius of the rod (r) = diameter / 2 = 13.00 mm / 2 = 6.50 mm = 0.0065 m

Now, let's calculate the moment of inertia of the disk:
I_disk = (1/2) * m * r^2
= (1/2) * 0.0160 kg * (0.0175 m)^2
≈ 2.429 * 10^-6 kg·m^2

Since the moment of inertia of the rod is neglected, the total moment of inertia (I_total) is equal to the moment of inertia of the disk (I_disk).

Next, let's calculate the change in angular velocity (Δω) using the formula Δω = (ωf - ωi), where ωf is the final angular velocity and ωi is the initial angular velocity.

Given that the initial angular velocity (ωi) is zero, we can write Δω = ωf.

Now, let's calculate ωf using the conservation of angular momentum:
L_initial = L_final
I_initial * ω_initial = I_final * ω_final
0 * ω_initial = I_total * ω_final (since the initial angular velocity is zero)
0 = (2.429 * 10^-6 kg·m^2) * ω_final

Solving for ω_final, we find that ω_final = 0 rad/s. (The top does not rotate as a result of pulling the string).

Therefore, the resulting angular velocity of the toy top is 0 rad/s (a).

To calculate the force exerted on the string, we can use Newton's second law: F = m * a, where F is the force, m is the mass, and a is the acceleration.

In this case, the mass of the toy top (m) is given as 0.0160 kg and the acceleration (a) can be found using the formula a = (v_f - v_i) / t, where v_f is the final velocity, v_i is the initial velocity, and t is the time taken.

Given that the initial velocity (v_i) is 0 m/s, the final velocity (v_f) is 3.0 m/s, and the time (t) is 0.40 s, we can calculate the acceleration:

a = (v_f - v_i) / t
= (3.0 m/s - 0 m/s) / 0.40 s
= 7.5 m/s^2

Now, we can calculate the force exerted on the string:

F = m * a
= 0.0160 kg * 7.5 m/s^2
= 0.12 N

Therefore, the force exerted on the string is 0.12 N (b).