A bench is made of a 3 meter long plank with a mass of 12kg and 2 sawhorses(each 0.2m from an end).There are two boxes on the bench. A 15kg box is 0.8 meter from the left hand side. A second box is 0.5 m from the right hand side. What would the mass of the second box have to be for the normal force exerted by the right sawhorse to be 3 times larger than that exerter by the left sawhorse?

Question

A bench is made of a 3 meter long plank with a mass of 12kg and 2 sawhorses(each 0.2m from an end).There are two boxes on the bench. A 15kg box is 0.8 meter from the left hand side. A second box is 0.5 m from the right hand side. What would the mass of the second box have to be for the normal force exerted by the right sawhorse to be 3 times larger than that exerter by the left sawhorse?

Answer 1

To find the mass of the second box that would result in the right sawhorse exerting a normal force three times larger than the left sawhorse, we can use the principle of torque.

First, let's calculate the torque exerted by each object on the plank about the left sawhorse. Torque is given by the formula:

Torque = force × perpendicular distance

For the left sawhorse, the torque is zero since the perpendicular distance is zero (0.2 meters from the left end).

For the right sawhorse, the torque can be calculated using the mass of the first box (15 kg) and the mass of the second box (m kg), along with their respective distances from the sawhorse:

Torque = (15 kg × g) × 0.2 m + (m kg × g) × (3 m - 0.5 m)

Where g is the acceleration due to gravity (9.8 m/s²).

Now, since we want the normal force exerted by the right sawhorse to be three times larger than the left sawhorse, we can equate the torques:

(15 kg × g) × 0.2 m + (m kg × g) × (3 m - 0.5 m) = 3 × (15 kg × g) × 0.2 m

Simplifying the equation:

3 × (15 kg × g) × 0.2 m = 3 × (15 kg × g) × 0.2 m - (m kg × g) × (2.5 m)

Now we can solve for the mass of the second box, m:

0 = 8 × (15 kg × g) × 0.2 m - (m kg × g) × (2.5 m)

Simplifying further:

0 = (120 kg × g) - (2.5 m × m × g)

Since we know g is 9.8 m/s², we can substitute that value:

0 = 1176 kg·m/s² - (2.5 m × m × 9.8 m/s²)

Simplifying again:

0 = 1176 - (24.5 m²)

Rearranging the equation:

24.5 m² = 1176

Solving for m²:

m² = 1176 / 24.5

m² ≈ 47.92

Taking the square root:

m ≈ √47.92

m ≈ 6.93

Therefore, the mass of the second box would need to be approximately 6.93 kg for the normal force exerted by the right sawhorse to be three times larger than that exerted by the left sawhorse.