A person stands on a bathroom scale in a motionless elevator begins to move,the scale briefly reads only 0.75 of th person regular weight(means Fn=0.75mg).Calculate the acceleration of the elevator,and find the direction of acceleration.
2.5 m/s2, down
To solve this problem, we can start by defining the forces acting on the person.
1. The weight of the person is given by W = mg, where m is the mass of the person and g is the acceleration due to gravity.
2. The normal force exerted by the scale is denoted by N. In this case, the scale reads only 0.75 of the person's regular weight, so N = 0.75mg.
Now, let's consider the forces acting on the person when the elevator starts to move:
1. The weight of the person, W = mg, acts downward.
2. The normal force, N = 0.75mg, acts upward.
3. The net force acting on the person is given by Newton's second law: ΣF = ma, where ΣF is the total force, m is the mass, and a is the acceleration of the elevator.
Since there are only two forces acting on the person in the vertical direction (weight and normal force), we can set up the equation:
ΣF = ma
N - W = ma
Substituting the given values for N and W:
0.75mg - mg = ma
Combining like terms:
0.75mg - mg = ma
-0.25mg = ma
Dividing both sides of the equation by m:
-0.25g = a
Therefore, the acceleration of the elevator is -0.25g (negative because it is directed opposite to the force of gravity).
To find the direction of acceleration, we need to consider the sign. Since the result is negative, the elevator is accelerating downward.
To calculate the acceleration of the elevator, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the person standing on the scale experiences a normal force (Fn) from the scale, which is less than their weight (mg) due to the reading being 0.75 times their regular weight. Let's denote the person's weight as W, so W = mg.
The net force acting on the person can be calculated as the difference between their weight and the normal force: Fnet = W - Fn.
Since the person is standing on the scale in a motionless elevator that begins to move, we know that the net force is equal to the mass of the person multiplied by their acceleration: Fnet = ma.
Equating these two expressions for the net force, we get:
W - Fn = ma
Substituting W = mg and Fn = 0.75mg:
mg - 0.75mg = ma
0.25mg = ma
Now, we can cancel out the mass (m) on both sides of the equation:
0.25g = a
So, the acceleration of the elevator is 0.25 times the acceleration due to gravity (g). If we know the value of g (which is approximately 9.8 m/s^2 on Earth), we can multiply it by 0.25 to find the numerical value of the acceleration.
As for the direction of acceleration, it depends on the context. If the elevator is moving upwards, the acceleration would be positive (+0.25g), indicating upward acceleration. If the elevator is moving downwards, the acceleration would be negative (-0.25g), indicating downward acceleration.