Chapter 4, Problem 37P

(a) An elevator of mass m moving upward has two forces acting on it: the upward force of tension in the cable and the downward force due to gravity. When the elevator is accelerating upward, which is greater, T or w? (b) When the elevator is moving at a constant velocity upward, which is greater, T or w? (c) When the elevator is moving upward, but the acceleration is downward, which is greater, T or w? (d) Let the elevator have a mass of 1 500 kg and an upward acceleration of 2.5 m/s². Find T. Is your answer consistent with the answer to part (a)? (e) The elevator of part (d) now moves with a constant upward velocity of 10 m/s. Find T. Is your answer consistent with your answer to part (b)? (f) Having initially moved upward with a constant velocity, the elevator begins to accelerate downward at 1.50 m/s². Find T. Is your answer consistent with your answer to part (c)?

To determine

The tension on the cable.

Answer to Problem 37P

Tension (T) on the cable should be greater than the weight.

Explanation of Solution

Given Info: Mass of the block is m.

Weight of the elevator is,

  $W=mg$

• g is the acceleration due to gravity.
• m is  the mass of the block.

Conclusion:

The force of gravity acts downwards. For the elevator to move upwards,$T>W$.

To determine

The tension on the cable.

Answer to Problem 37P

Tension (T) on the cable is equal to the weight.

Explanation of Solution

According to Newton’s second law, force is equal to the product of mass and acceleration.

Force is expressed as,

  $F=ma$

• a is the acceleration.
• m is  the mass of the block.

Conclusion:

Acceleration is the rate of change of velocity. Since, velocity is constant, the acceleration is zero. Therefore, the total force is zero. As a result, the tension equals the weight of the elevator.

To determine

The tension on the cable.

Answer to Problem 37P

Tension (T) on the cable should be lesser than the weight.

Explanation of Solution

Given Info: Mass of the block is m.

Weight of the elevator is,

  $W=mg$

• g is the acceleration due to gravity.
• m is  the mass of the block.

Conclusion:

The force of gravity acts downwards. For the elevator to move downwards, $T<W$.

To determine

The tension on the cable.

Answer to Problem 37P

Tension (T) on the cable should be greater than the weight.

Explanation of Solution

Tension on the cable is,

  $T=m\left(g+a\right)$

• g is the acceleration due to gravity.
• m is  the mass of the block.
• a is the acceleration.

Weight of the elevator is,

  $W=mg$

• g is the acceleration due to gravity.
• m is  the mass of the block.

Substitute 1200 kg for m, $9.8{\text{ms}}^{-2}$ for g and $2.50{\text{ms}}^{-2}$ for a in the expression for T.

$\begin{array}{c}T=\left(1200\text{kg}\right)\left(9.8{\text{ms}}^{-2}+2.50{\text{ms}}^{-2}\right)\\ =14760\text{N}\end{array}$

Substitute 1200 kg for m and $9.8{\text{ms}}^{-2}$ for g in the expression for W.

$\begin{array}{c}W=\left(1200\text{kg}\right)\left(9.8{\text{ms}}^{-2}\right)\\ =11760\text{N}\end{array}$

Conclusion:

Tension on the cable is 14760 N. The weight of the elevator is 11760 N. Therefore,

T > W.

To determine

The tension on the cable.

Answer to Problem 37P

Tension (T) on the cable is equal to the weight.

Explanation of Solution

Tension on the cable is,

  $T=m\left(g+a\right)$

• g is the acceleration due to gravity.
• m is  the mass of the block.
• a is the acceleration.

Acceleration is the rate of change of velocity. Since, velocity is constant, the acceleration is zero

Weight of the elevator is,

  $W=mg$

• g is the acceleration due to gravity.
• m is  the mass of the block.

Substitute 1200 kg for m, $9.8{\text{ms}}^{-2}$ for g and $0{\text{ms}}^{-2}$ for a in the expression for T.

$\begin{array}{c}T=\left(1200\text{kg}\right)\left(9.8{\text{ms}}^{-2}+0{\text{ms}}^{-2}\right)\\ =11760\text{N}\end{array}$

Substitute 1200 kg for m and $9.8{\text{ms}}^{-2}$ for g in the expression for W.

$\begin{array}{c}T=\left(1200\text{kg}\right)\left(9.8{\text{ms}}^{-2}\right)\\ =11760\text{N}\end{array}$

Conclusion:

Therefore, T = W. It is consistent with (b).

To determine

The tension on the cable.

Answer to Problem 37P

Tension (T) on the cable should be lesser than the weight.

Explanation of Solution

Tension on the cable is,

  $T=m\left(g-a\right)$

• g is the acceleration due to gravity.
• m is  the mass of the block.
• a is the acceleration.

Weight of the elevator is,

  $W=mg$

• g is the acceleration due to gravity.
• m is  the mass of the block.

Substitute 1200 kg for m, $9.8{\text{ms}}^{-2}$ for g and $2.50{\text{ms}}^{-2}$ for a in the expression for T.

$\begin{array}{c}T=\left(1200\text{kg}\right)\left(9.8{\text{ms}}^{-2}-2.50{\text{ms}}^{-2}\right)\\ =8760\text{N}\end{array}$

Substitute 1200 kg for m and $9.8{\text{ms}}^{-2}$ for g in the expression for W.

$\begin{array}{c}T=\left(1200\text{kg}\right)\left(9.8{\text{ms}}^{-2}\right)\\ =11760\text{N}\end{array}$

Conclusion:

Therefore, $T<W$. It is consistent with (c).