Final answer:
It is derived by applying the equations for circular motion, specifically using cosine and sine functions for the x and y positions, respectively. The position of the particle in circular motion with an angular frequency of 4.00 rad/s is at x = 5 m, y = 0 m after 10 seconds, which corresponds to option (b).
Step-by-step explanation:
The subject question involves a particle executing circular motion with a constant angular frequency. The position of the particle at a specific time can be found using the formulas for circular motion in physics. When the angular frequency (ω) is 4.00 rad/s and the particle starts from x = 5 m, y = 0 m at t = 0, the position at t = 10 s can be determined using the equations of motion for uniform circular motion:
- The x-position is given by x(t) = R × cos(ωt), where R is the radius of the circular path.
- The y-position is given by y(t) = R × sin(ωt).
The radius R can be deduced from the initial conditions, which is 5 m in this case. Plugging in the values, we get:
- x(10) = 5 × cos(40) = 5 × cos(2π × 10/10) = 5 × cos(2π) = 5 m
- y(10) = 5 × sin(40) = 5 × sin(2π × 10/10) = 5 × sin(2π) = 0 m
Thus, the position of the particle at t = 10 s is x = 5 m, y = 0 m, which corresponds to option (b). We have used the fact that the function of cosine is periodic, and its value repeats every 2π radians, leading to a cos(2π) = 1.
To find the velocity, we would differentiate the position functions with respect to time. To determine the acceleration, especially the centripetal acceleration, we would differentiate the velocity function or use the formula ac = -ω2R.