A 62kg person lies flat on uniform plank of mass 15kg.The plank, with the person lying on it,is placed on a brick at the head and a bathroom scale at the foot,both under the plank.The persons toe to head dist is 1.56m.The length of the plank is also 1.56m.
a)the reading on the bathroom scale is 30kg.Use this information to determine how far the centre of gravity of the person is from the toes.
30*1.56=46.8
15*0.78=11.7
46.8-11.7=35.1
35.1/62=0.567
Answer = 0.567 meters from pivot which is = to (1.56-0.567) 0.993m from the scales
To determine how far the center of gravity of the person is from the toes, we can use the concept of moments or torques.
First, let's calculate the weight of the person, which is equal to their mass multiplied by the acceleration due to gravity (9.8 m/s^2):
Weight of the person = mass × gravitational acceleration
= 62 kg × 9.8 m/s^2
= 607.6 N
Now, let's consider the torques acting on the plank. Torques are calculated by multiplying the force applied by the perpendicular distance from the pivot point (in this case, the brick) to the line of action of the force.
The torques acting clockwise and anticlockwise should balance each other, which means their magnitudes would be equal.
Clockwise torque: 15 kg × gravitational acceleration × distance (to be determined)
Anticlockwise torque: 30 kg × gravitational acceleration × 1.56 m (length of the plank)
Since the torques are equal:
15 kg × gravitational acceleration × distance = 30 kg × gravitational acceleration × 1.56 m
Cancelling out the gravitational acceleration from both sides of the equation, we get:
15 kg × distance = 30 kg × 1.56 m
Simplifying the equation further:
distance = (30 kg × 1.56 m) / 15 kg
Evaluating the expression:
distance = 3.12 m / 15 kg
distance = 0.208 m
Therefore, the center of gravity of the person is approximately 0.208 meters (or 208 millimeters) from their toes.
To determine how far the center of gravity of the person is from the toes, we can utilize the concept of torques. Torque is defined as the product of the force and the perpendicular distance from the point of rotation.
In this case, the point of rotation is the brick at the head. The force acting on the plank-person system is the weight of the system, which is the sum of the person's weight and the plank's weight.
The equation for torque is:
Torque = Force × Distance
Let's calculate the torque caused by the person:
Weight of the person = mass of the person × acceleration due to gravity
= 62 kg × 9.8 m/s²
= 607.6 N
The distance from the person's weight to the brick is half the length of the plank:
Distance = 1.56 m ÷ 2 = 0.78 m
Torque caused by the person = 607.6 N × 0.78 m
= 474.408 N·m
Now, let's calculate the torque caused by the plank:
Weight of the plank = mass of the plank × acceleration due to gravity
= 15 kg × 9.8 m/s²
= 147 N
The distance from the plank's weight to the brick is the full length of the plank:
Distance = 1.56 m
Torque caused by the plank = 147 N × 1.56 m
= 229.32 N·m
The sum of the torques caused by the person and the plank is the total torque acting on the system. Since the system is in equilibrium, this total torque should be zero.
Total torque = Torque caused by the person + Torque caused by the plank
= 0
Therefore,
474.408 N·m + 229.32 N·m + Torque caused by the bathroom scale = 0
Torque caused by the bathroom scale = -703.728 N·m
Now, we can calculate the distance from the center of gravity of the system to the toes:
Distance from the center of gravity to the toes = |Torque caused by the bathroom scale| ÷ Weight of the person
Distance from the center of gravity to the toes = 703.728 N·m ÷ (62 kg × 9.8 m/s²)
≈ 1.16 m
Therefore, the center of gravity of the person is approximately 1.16 meters from the toes.