The upper surface of a cube of gelatin, 6.9 cm on a side, is displaced 0.67 cm by a tangential force. If the shear modulus of the gelatin is 946 Pa, what is the magnitude of the tangential force?
To find the magnitude of the tangential force applied to the gelatin cube, we can use Hooke's Law and the formula for shear stress.
Hooke's Law states that the force applied to an object is directly proportional to the deformation or displacement of the object. In this case, the displacement of the gelatin cube is 0.67 cm.
The formula for shear stress is:
shear stress = (Shear modulus) × (shear strain)
In this case, the shear modulus of the gelatin is given as 946 Pa, and we need to find the shear strain.
Shear strain is defined as the ratio of the displacement perpendicular to the applied force to the original length of the object. Since the cube is displaced in only one direction (tangential to the surface), the shear strain is equal to the displacement.
Therefore, the shear strain is 0.67 cm or 0.0067 m.
Now, we can substitute these values into the formula for shear stress to find the magnitude of the tangential force.
shear stress = (Shear modulus) × (shear strain)
shear stress = 946 Pa × 0.0067 m
Calculating this expression gives us the shear stress in Pascals (Pa).
Finally, we can convert the shear stress from Pascals to Newtons by multiplying it by the area over which the force is applied. In this case, the area is the upper surface of the cube, which is a square with sides of length 6.9 cm. The area is calculated as:
area = side length × side length
area = 6.9 cm × 6.9 cm
Then, convert the area from square centimeters to square meters by dividing by 10,000 (since there are 10,000 square centimeters in a square meter).
Now, we can multiply the shear stress by the area to find the magnitude of the tangential force:
tangential force = shear stress × area
Plugging in the values, we can calculate the magnitude of the tangential force.