A force, F = (3x + 4y) N,
is applied to an object at a point whose position vector with respect to the pivot point is
r = (3x + 3y + 3z) m.
Calculate the torque created by the force about that pivot point.
I know that torque = -12x + 9y + 3z, but why is -12x negative? Thank you in advance.
Because Torque is a cross product of radius vector(r) and Force vector(F).
If r =Ax+By+Cz & F = ax+by+cz (where x,y and z are unit vectors in x,y and z directions) then
x-component of torque Tx is :
Tx = (Bc-Cb)
= 3*0-3*4
= -12
Therefore x-comp. of the Torque is nagative (-12)
To calculate the torque created by a force about a pivot point, we use the formula:
Torque = r x F
where "x" represents the cross product.
In this case, we have the force F = (3x + 4y) N, and the position vector r = (3x + 3y + 3z) m.
To compute the cross product r x F, we need to determine the components of the resulting vector by taking the determinants of the following matrix:
| i j k |
| 3x 3y 3z |
| 3x+4y 3y+4x 0 |
Expanding the determinants, we can perform the calculations as follows:
i-component = (3y)(0) - (3z)(3y+4x)
j-component = (3z)(3x+4y) - (3x)(0)
k-component = (3x)(3y+4x) - (3y)(3x)
Simplifying these calculations, we get:
i-component = -12xy - 9xz
j-component = 9xz + 12yz
k-component = 9xy - 9yx
Combining the i, j, and k components, we obtain the torque vector as:
Torque = (-12xy - 9xz)i + (9xz + 12yz)j + (9xy - 9yx)k
Now, looking specifically at the x-component of the torque, we have -12xy. Notice that the x-component of the torque is indeed negative (-12), indicating that it creates a clockwise torque or moment about the pivot point. The negative value arises because the force F is applied in the positive x-direction (3x) and the position vector r has both x and y components. The negative x-component signifies the opposite rotational direction to the force.
Hence, the negative sign in -12x indicates the direction of the torque relative to the force and position vector.
Hope this clarifies your query! Let me know if you need any further assistance.