The coordinates of an object moving in the xy plane vary with time according to the following equations x = −7.16 sin ùt and y = 4.00 − 7.16 cos ùt, where ù is a constant, x and y are in meters, and t is in seconds. Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t > 0. (Use the following variables as necessary: omega for ù and t.)
For the position vector I got:
-7.16sin(omega*t)i+4.00-7.16cos(omega*t)j
For the velocity vector I got:
-7.16*omega*cos(omega*t)i+7.16*omega*sin(omega*t)j
For the acceleration vecotr I got:
7.16*omega^2*sin(omega*ti)+7.16*omega^2*sin(omega*t)j
I this correct or not? My homework is computer based and when I type this in, it marks it wrong. Please help.
The only two things I see wrong is you have on acceleration the unit vector i inside the argument of the sin function, and on the lst part of the acceleration vector, you should have COSINE, not sine
One error
For the acceleration vector I got:
7.16*omega^2*sin(omega*ti)+7.16*omega^2*__________sin(omega*t)______j
should be
For the acceleration vector I got:
7.16*omega^2*sin(omega*ti)+7.16*omega^2*cos(omega*t)j
Note that
acceleration = - omega^2 *( sin and cos components of displacement)
Your expressions for the position vector, velocity vector, and acceleration vector are correct. However, it's possible that the computer-based system is looking for a different format or notation. Here's an alternative way to express the vectors:
Position vector: r(t) = -7.16 sin(omega*t) i + (4.00 - 7.16 cos(omega*t)) j
Velocity vector: v(t) = -7.16 omega cos(omega*t) i + 7.16 omega sin(omega*t) j
Acceleration vector: a(t) = -7.16 omega^2 sin(omega*t) i - 7.16 omega^2 cos(omega*t) j
Make sure you are using the correct value for the constant omega, and check for any additional formatting requirements in your homework system.
Your expressions for the position vector, velocity vector, and acceleration vector are mostly correct. However, there is a small mistake in the expression for the acceleration vector. Let's go through each of them to identify the mistake.
Position Vector:
Given that x = -7.16 sin(ωt) and y = 4.00 - 7.16 cos(ωt), the position vector can be written as:
r = x i + y j = (-7.16 sin(ωt)) i + (4.00 - 7.16 cos(ωt)) j
So, your expression for the position vector is correct:
-7.16 sin(ωt) i + (4.00 - 7.16 cos(ωt)) j
Velocity Vector:
The velocity vector is the derivative of the position vector with respect to time (t). Taking the derivatives of x and y with respect to t, we get:
vx = dx/dt = d/dt(-7.16 sin(ωt)) = -7.16ω cos(ωt)
vy = dy/dt = d/dt(4.00 - 7.16 cos(ωt)) = 7.16ω sin(ωt)
So, the velocity vector can be expressed as:
v = vx i + vy j = (-7.16ω cos(ωt)) i + (7.16ω sin(ωt)) j
Your expression for the velocity vector is correct:
-7.16ω cos(ωt) i + 7.16ω sin(ωt) j
Now, let's check the expression for the acceleration vector.
Acceleration Vector:
The acceleration vector is the derivative of the velocity vector with respect to time (t). Taking the derivatives of vx and vy with respect to t, we get:
ax = d²x/dt² = d/dt(-7.16ω cos(ωt)) = 7.16ω² sin(ωt)
ay = d²y/dt² = d/dt(7.16ω sin(ωt)) = 7.16ω² cos(ωt)
So, the acceleration vector can be expressed as:
a = ax i + ay j = (7.16ω² sin(ωt)) i + (7.16ω² cos(ωt)) j
There was a small mistake in your expression for the acceleration vector. The correct expression for the acceleration vector is:
7.16ω² sin(ωt) i + 7.16ω² cos(ωt) j
Please recheck your homework submission and ensure that your expressions match the correct forms mentioned above.