Hey everyone, i am currently stuck on a physics problem. It goes like this: Assume there is a particle goes in a circle with an angle velocity ω=at^2+b(function of time), which …

Question

Hey everyone, i am currently stuck on a physics problem.

It goes like this:

Assume there is a particle goes in a circle with an angle velocity ω
`=at^2+b`(function of time), which a = 0.01 (radian/
`s^(2)`) and b = 0.05 (radian/s). Find the angle between vector of net acceleration and vector of tangential velocity at t=2s.


(It has driven me nuts not gonna lie, because it's not uniform circular motion or constant-acceleration motion)

Answer 1

To find the angle between the vector of net acceleration and the vector of tangential velocity at t = 2 seconds, we need to first calculate the net acceleration and tangential velocity at that specific time.

Given:

ω = dθ/dt = αt^2 + b (angular velocity as a function of time)

where α = 0.01 rad/s^2 and b = 0.05 rad/s.

To find the angle between the vector of net acceleration and the vector of tangential velocity, we'll use the following steps:

Step 1: Calculate the angular displacement (θ) at t = 2 seconds.

Step 2: Calculate the tangential velocity (v_t) at t = 2 seconds.

Step 3: Calculate the net acceleration (a_net) at t = 2 seconds.

Step 4: Find the angle (θ_angle) between the vectors of net acceleration and tangential velocity.

Let's proceed with the calculations:

Step 1: Calculate the angular displacement (θ) at t = 2 seconds.

θ = ∫(ω) dt = ∫(αt^2 + b) dt

θ = α * (t^3 / 3) + b * t + C

Since we don't have the constant of integration (C), we'll use initial conditions to find it. At t = 0 seconds, θ = 0 (starting from the initial position). Therefore, C = 0.

θ = α * (t^3 / 3) + b * t

At t = 2 seconds:

θ = 0.01 * (2^3 / 3) + 0.05 * 2

θ ≈ 0.01 * (8 / 3) + 0.1

θ ≈ 0.0267 + 0.1

θ ≈ 0.1267 radians

Step 2: Calculate the tangential velocity (v_t) at t = 2 seconds.

v_t = ω * r

Where r is the radius of the circular path. However, the radius is not given in the problem. We need more information about the circular motion (e.g., radius or the position of the particle) to calculate the tangential velocity.

Step 3: Calculate the net acceleration (a_net) at t = 2 seconds.

a_net = dω/dt = α * 2t = 0.01 * 2 * 2 = 0.04 rad/s^2

Step 4: Find the angle (θ_angle) between the vectors of net acceleration and tangential velocity.

Without the tangential velocity (v_t), we cannot directly calculate the angle between the vectors of net acceleration and tangential velocity.

If you provide additional information about the radius or position of the particle, I can help you calculate the tangential velocity and the angle between the vectors.