A small object with mass 4.05 kg moves counterclockwise with constant speed 1.70 rad/s in a circle of radius 3.15 m centered at the origin. It starts at the point with position vector 3.15 m. Then it undergoes an angular displacement of 9.10 rad.
(a) What is its new position vector?
r cos(angular displacement) =i
r sin(angular displacement) =j
3.15m * cos(9.10 rad) = -2.98
3.15m * sin(9.10 rad) = 1.01
(-2.98i+1.01j)m
To find the new position vector of the object after undergoing an angular displacement, we can use the formula:
r = R * cos(theta) + R * sin(theta)
where:
- r is the new position vector
- R is the radius of the circle
- theta is the new angular displacement
Given:
- Radius (R): 3.15 m
- Angular displacement (theta): 9.10 rad
Substituting these values into the formula, we can calculate the new position vector:
r = 3.15 * cos(9.10) + 3.15 * sin(9.10)
Using a calculator, we find:
r ≈ -1.701 m
Therefore, the new position vector of the object is approximately -1.701 m.
To find the new position vector of the object, we need to use the concept of angular displacement and the formula for finding the position vector in circular motion.
We know that the object starts at a point with a position vector of 3.15 m, and it undergoes an angular displacement of 9.10 rad. The object is moving counterclockwise, so the angular displacement is positive.
The formula for finding the position vector in circular motion is:
r = R * cos(θ)i + R * sin(θ)j
where r is the position vector, R is the radius of the circle, θ is the angular displacement, i and j are unit vectors in the x and y directions, respectively.
Substituting the given values into the formula, we have:
r = 3.15 m * cos(9.10 rad)i + 3.15 m * sin(9.10 rad)j
Using a calculator, we can evaluate the cosine and sine of 9.10 rad:
cos(9.10 rad) ≈ -0.0349
sin(9.10 rad) ≈ 0.9994
Substituting these values back into the formula, we get:
r ≈ -0.0349 * 3.15 m * i + 0.9994 * 3.15 m * j
Simplifying, we find:
r ≈ -0.1098i + 3.1517j
Therefore, the new position vector of the object is approximately (-0.1098, 3.1517) meters.