Two scales support a massless plank from either end, separated by 3 m. A person, whose height is less than 3 m, lies down on the plank with their head on the leftmost scale. The leftmost scale reads 314 N, and the rightmost scale reads 112 N.

(a)
Calculate the person’s mass, in kg.
(b)
What is the location, in metres from the person’s head, of their centre of mass?

To solve this problem, we can use the principle of moments or torque. Torque is the product of the force applied and the perpendicular distance from the point of rotation (or pivot). In this case, the pivot is the leftmost scale, and we need to find both the person's mass and the location of their center of mass.

(a) Calculating the person's mass:

1. Start by converting the forces on the scales to their respective torques. Torque is given by the formula T = F x d, where T is the torque, F is the force, and d is the distance.

Leftmost scale torque = 314 N x 3 m = 942 Nm
Rightmost scale torque = 112 N x 0 m (since the person's head is at the pivot point) = 0 Nm

2. Since the person is lying down, we need to consider their body as a uniform distribution of mass along the plank. Let's assume the person's mass is m kg.

Next, we need to find the distance from the person's head to their center of mass. Let's assume this distance is x meters.

3. The torque equation for the system can be set up as follows:
Leftmost scale torque = Rightmost scale torque

314 N x 3 m = m kg x g m/s^2 x x m
942 Nm = m kg x 9.8 m/s^2 x x m

4. We can rearrange the equation to solve for m:
m kg = 942 Nm / (9.8 m/s^2 x x m)

(b) Calculating the location of the person's center of mass:

1. We can use the equation for the center of mass:
center of mass = (Leftmost scale torque) / (Force on the leftmost scale)

center of mass = 942 Nm / 314 N

2. Simplifying the equation gives us:
center of mass = 3 m

Therefore, the person's mass is given by m kg, and the location of their center of mass is x = 3 m from their head.

To solve this problem, we can use the principles of torque and center of mass. First, let's find the answers to part (a) and (b) step by step:

(a) Calculate the person's mass, in kg:
To find the person's mass, we can use the equation: Weight = Mass x Gravity
The weight is the force measured by the scales, which is 314 N on the left side.
The gravity, denoted by g, is approximately 9.8 m/s².

Using the equation, we can solve for the person's mass:
314 N = Mass x 9.8 m/s²

Rearranging the equation, we have:
Mass = 314 N / 9.8 m/s² ≈ 32 kg

Therefore, the person's mass is approximately 32 kg.

(b) What is the location, in meters from the person's head, of their center of mass:
To find the location of the person's center of mass, we need to consider the torques acting on the plank.
The torque is equal to the force applied multiplied by the distance from the rotation point (fulcrum).

In this case, the person's head is the rotation point (fulcrum), so there is no torque contribution from the head.
The torques acting on the plank are as follows:
- The weight of the person, which acts downwards at the center of mass.
- The normal force from the left scale, which acts upwards at a distance of 3 meters from the person's head.
- The normal force from the right scale, which also acts upwards at an unknown distance from the person's head.

For equilibrium, the torques on both sides must balance out.
The torque is calculated as Torque = Force x Distance.

Given that the left scale reads 314 N at a distance of 3 m, and the right scale reads 112 N at an unknown distance (denoted as x), we can set up the equation:

(314 N × 3 m) = (112 N × x)

Rearranging the equation, we have:
x = (314 N × 3 m) / 112 N
x ≈ 8.39 m

Therefore, the location of the person's center of mass is approximately 8.39 meters from their head.

A spiral spring of nature 20cm,has a scale pan hanging freely in its lower end. When an object of mass 40g is placed in the pan,its length becomes 21.8cm.when another object of mass 60g is placed in the pan,the length becomes 22.05cm. Calculate the mass of the scale pan