A lamp hangs vertically from a cord in a descending elevator that declerates at 2.4m/s^2 a) if the tension in the cord is 89N what is the lamp's mass? b) what is the cord's tension when the elevator is ascends with an upward acceleration of 2.4m/s^2?

I know T=w where w=mg. Would I use T-mg=ma? I should be getting 31.3 kN for a) and 24.3kN for part b but I can't figure out my problem.

Yes, use
T - mg = ma.
T does NOT equal mg when the elevator is accelerating or decelerating.

In the first case,
89 N = m (g + a)
m = 89/(9.8 + 2.4) = 7.3 kg
I used a plus sign for "a" since it is decelerating and descending

The (a) problem asks for a mass, so I don't see how the answer can be in Newtons.

I don't agree with your part (b) "should" answer either.

Acceleration used should be -2.4.

If you used acceleration as +2.4, the tension should be negative according to its free diagram
mg-t=ma , which yields to 12.02 kg

For part (b), when the elevator is ascending with an upward acceleration, the tension in the cord will be the sum of the force due to gravity and the force due to the acceleration.

Using the equation T - mg = ma, we can rearrange it to solve for T:

T = mg + ma,

where T is the tension, m is the mass of the lamp, g is the acceleration due to gravity, and a is the acceleration of the elevator.

In this case, the acceleration is 2.4 m/s^2 (upward) and the acceleration due to gravity is 9.8 m/s^2 (downward).

Substituting the given values, we have:

T = m(9.8) + m(2.4),
T = 12.2m.

However, we don't have the mass of the lamp given in the problem. So we can't calculate the exact tension in the cord without knowing the lamp's mass.

To find the mass of the lamp in part (a), you correctly used the equation T - mg = ma, where T is the tension in the cord, m is the mass of the lamp, g is the acceleration due to gravity, and a is the acceleration of the elevator.

In this case, the elevator is decelerating at 2.4 m/s^2, so the acceleration a is negative (-2.4 m/s^2). The tension in the cord is given as 89 N.

Setting up the equation, we have 89 N - mg = ma. Rearranging the equation, we get -mg = ma - 89 N. Factoring out m, we have -m(g + a) = -89 N.

Now we can solve for the mass m. Divide both sides of the equation by -(g + a): m = 89 N / (g + a). Plugging in the values, we have m = 89 N / (9.8 m/s^2 - 2.4 m/s^2) = 7.3 kg.

So, the mass of the lamp in part (a) is indeed 7.3 kg, not 31.3 kN.

For part (b), we can use the same equation T - mg = ma, with the elevator now ascending with an upward acceleration of 2.4 m/s^2. Again, T is the tension in the cord, m is the mass of the lamp, g is the acceleration due to gravity, and a is the acceleration of the elevator.

Plugging in the values, we have T - mg = ma. Since the elevator is ascending, the acceleration a is positive (2.4 m/s^2). We need to solve for T, the tension in the cord.

Rearranging the equation, we get T = ma + mg. Plugging in the values, we have T = (m * 2.4 m/s^2) + (m * 9.8 m/s^2). Factoring out m, we have T = m * (2.4 m/s^2 + 9.8 m/s^2).

To find the cord's tension, we need the mass of the lamp. Using the previous result from part (a), the mass of the lamp is 7.3 kg. Plugging in this value, we have T = 7.3 kg * (2.4 m/s^2 + 9.8 m/s^2) = 82.6 N.

Therefore, the tension in the cord when the elevator is ascending with an upward acceleration of 2.4 m/s^2 is 82.6 N, not 24.3 kN.