Question

A ball of mass M is suspended by a thin string (of negligible mass) from the ceiling of an elevator. The vertical motion of the elevator as it travels up and down is described in the statements below. Indicate for each of the situations described the relation between value of the tension in the cable, T, and the weight of the ball, Mg, or whether one Cannot tell. (Assume that there is no air, i.e., neglect the buoyancy effect of the air.)T > Mg, T < Mg, T = Mg​, cannot tell

1.The elevator is traveling upward and its upward velocity is increasing as it begins its journey towards a higher floor.

2. The elevator is traveling upward at a constant velocity.

3.The elevator is traveling downward and its downward velocity is decreasing as it nears a stop at a lower floor.

4. The elevator is traveling upward and its upward velocity is decreasing as it nears a stop at a higher floor.

5.The elevator is traveling downward and its downnward velocity is increasing

6. The elevator is stationary and remains at rest.

Answer 1

We assume upward direction as positive direction. If accelration turns out to be in downward direction, we will take it as negative
In the free body Diagram above, using net force = mass * acceleration, we get
T-Mg=Ma
Or T= Mg + Ma
If we use the acceleration as positive or negative in the above equation, we can deduce whether T>Mg or T<Mg

(1) since upward velocity is increasing, acceleration is upwards and hence positive
So, T= Mg + Ma, Therefore T>Mg

(2) For constant velocity, acceleration is zero
So, T= Mg + M*0, Therefore T=Mg

(3) since downward velocity is decreasing, acceleration is upwards and hence positive
So, T= Mg + Ma, Therefore T>Mg

(4) since upward velocity is decreasing, acceleration is downwards and hence negative
So, T= Mg + M(-a)= Mg - Ma, Therefore T>Mg

(5) since downward velocity is increasing, acceleration is downwards and hence negative
So, T= Mg + M(-a)= Mg - Ma, Therefore T>Mg

(6) Here elevator is at rest i.e. its velocity is and stays zero. For constant velocity, acceleration is zero
So, T= Mg + M*0, Therefore T=Mg