In the design of a control mechanism, the vertical slotted guide is moving with a constant velocity x'= 15in/s during the interval of motion from
x= -8in to x= +8in. For the instant when x= 6in calculate the n- and t-components of acceleration of the pin P, which is confined to move in the parabolic slot. From these results, determine the radius of curvature p of the path at this position. I have the picture if you need to see it but it won't let me post the link on here.
I figured out that the equation for the parabola is y= 10 -(x^2)/10 and that the velocity in the y is -18in/s meaning the total velocity is 23.43in/s. How do I get the n- and t- components of the acceleration from this?
The n- and t- components of the acceleration can be calculated using the equation a = v' + v^2/r, where v' is the change in velocity and v is the velocity. In this case, the n-component of the acceleration is a = -18in/s + (23.43in/s)^2/r, and the t-component of the acceleration is a = 0 + (23.43in/s)^2/r. The radius of curvature p can then be calculated using the equation p = v^2/a, where a is the acceleration. Therefore, the radius of curvature p is equal to (23.43in/s)^2/a, where a is the acceleration.
To find the normal (n) and tangential (t) components of acceleration, we can start by expressing the position of the pin P on the parabolic slot as a function of time. Since the vertical slotted guide is moving with a constant velocity of 15 in/s, the horizontal position of the pin P can be given as x = 15t.
To determine the time at which x = 6 in, we can set up the following equation:
15t = 6
Solving for t, we find:
t = 6/15 = 0.4 s
Now we can differentiate the equation of the parabola, y = 10 - (x^2)/10, with respect to time, t, to obtain the velocity components. Since the y-component of velocity is -18 in/s, we have:
y' = -18 in/s
Differentiating the equation of the parabola with respect to x, we have:
(dy/dx) = (d(10 - (x^2)/10)/dx)
=> (dy/dx) = -x/5
To find dy/dt (the rate of change of y with respect to time), we can use the chain rule:
(dy/dt) = (dy/dx) * (dx/dt)
=> (dy/dt) = (-x/5) * 15
At x = 6 in (which corresponds to t = 0.4 s), we can calculate (dy/dt) by substituting the values:
(dy/dt) = (-6/5) * 15
=> (dy/dt) = -18 in/s
Now, we have the values of (dx/dt) = 15 in/s and (dy/dt) = -18 in/s. These represent the tangential and normal components of velocity respectively.
To find the n- and t- components of acceleration, we can differentiate the velocity components with respect to time. The tangential acceleration (at) represents the rate of change of tangential velocity, and the normal acceleration (an) represents the rate of change of normal velocity.
To obtain at, we differentiate (dx/dt) with respect to time:
at = (d(dx/dt)/dt) = (d^2x/dt^2)
Since (dx/dt) is constant at 15 in/s, its rate of change is zero:
at = 0
To find an, we differentiate (dy/dt) with respect to time:
an = (d(dy/dt)/dt) = (d^2y/dt^2)
Differentiating (-18) with respect to t, we again find that an is zero:
an = 0
Therefore, at this position on the parabolic slot, the normal (n) and tangential (t) components of acceleration are both zero.
Now, to determine the radius of curvature (p) of the path, we can use the formula:
p = (v^2) / a
where v represents the magnitude of velocity and a represents the magnitude of acceleration. In this case, both v and a are zero, so the radius of curvature (p) is undefined or infinite.
Note: If there are any additional information or details in the given problem statement or the provided picture, it may be helpful to provide them for a more accurate analysis and calculation.
To calculate the n- and t-components of acceleration for the given scenario, you can use the following steps:
1. Find the derivatives of the position equation with respect to time to obtain the velocity equation and acceleration equation.
Given position equation: y = 10 - (x^2)/10
First, let's find the velocity equation, v(t), by differentiating the position equation with respect to time:
dx/dt = x' = 15 in/s (constant velocity)
dy/dt = d(10 - (x^2)/10)/dt
= -x/5 * (dx/dt)
= -3x in/s
So, the velocity equation is: v = v_x * i + v_y * j
where v_x = 15 in/s (given) and v_y = -3x in/s.
2. To find the acceleration equation, differentiate the velocity equation with respect to time:
d^2x/dt^2 = a_x = 0 (since x' is constant)
d^2y/dt^2 = d(-3x)/dt
= -3(dx/dt)
= -3 * 15
= -45 in/s^2
So, the acceleration equation is: a = a_x * i + a_y * j
where a_x = 0 and a_y = -45 in/s^2.
3. To find the n- and t-components of acceleration, we need to resolve the acceleration vector a into its normal (n) and tangential (t) components.
The normal component of acceleration (a_n) represents the acceleration perpendicular to the path, while the tangential component of acceleration (a_t) represents the acceleration along the path.
At any instant, the acceleration vector a can be written as:
a = a_n * n + a_t * t
Where n is the unit vector in the direction of the normal and t is the unit vector in the direction of the tangent.
To find the n- and t-components, you need to determine the unit vectors n and t at the given position.
4. To find the unit vectors n and t, we use the derivatives of the position equation.
For the given position equation: y = 10 - (x^2)/10
We know that the tangent vector t is given by:
t = (dx/dt)i + (dy/dt)j
Substituting the given values of dx/dt and dy/dt (15 and -3x, respectively) at x = 6in, we get:
t = (15)i - (18)j
To find the normal vector n, we have:
n = -dy/dt i + dx/dt j
Substituting the given values of dx/dt and dy/dt (15 and -3x, respectively) at x = 6in, we get:
n = (18)i + (15)j
5. Finally, with the known values of a, n, and t, we can find the n- and t-components of acceleration by taking their dot products with the respective unit vectors:
a_n = a • n
= (a_x)i + (a_y)j • (18)i + (15)j
a_t = a • t
= (a_x)i + (a_y)j • (15)i - (18)j
6. By calculating a_n and a_t, you can determine the radius of curvature p using the relation:
p = (v^2)/a_n
Substitute the calculated values of v (total velocity) and a_n into the equation to find the radius of curvature at the given position.