U= -30
m = 1/2

we know that m = -v/u
so v = 15

1/f = 1/v + 1/u 
 
f = 10

for m= 1/3 

-v/u = 1/3
v= -(1/3 * u)

1/f = 1/v + 1/u

1/10 = 3/-u + 1/u 

1/10 = -4 /u 

u = -40 

therefore for the magnification to be 1/3 the object must be placed 40 cm in front of the mirror

That's all

**Introduction**

To determine the position where an object should be placed to achieve a specific magnification with a concave spherical mirror, we will use the mirror formula and the magnification formula. The mirror formula is given as:

1/f = 1/v - 1/u

Where:
- f is the focal length of the mirror,
- v is the image distance from the mirror, and
- u is the object distance from the mirror.

The magnification formula is given as:

magnification (m) = -v/u

In this case, we are given that the magnification produced is 1/2 when the object is placed at a distance of 40 cm from the concave spherical mirror. We need to find the new position where the object should be placed to achieve a magnification of 1/3.

**Given Information**

- Object distance (u) = 40 cm
- Magnification (m) = 1/2

**Calculating the Focal Length**

First, we need to calculate the focal length (f) of the concave spherical mirror. We can rearrange the magnification formula to solve for v:

magnification (m) = -v/u -> v = -m * u

Substituting the given values:

v = - (1/2) * 40 cm = -20 cm

Now, we can substitute the values of v and u into the mirror formula and solve for f:

1/f = 1/v - 1/u

1/f = 1/(-20 cm) - 1/(40 cm)

1/f = -1/(20 cm)

f = -20 cm

The negative sign indicates that the mirror is concave.

**Calculating the New Object Distance**

Now, we need to find the new position where the object should be placed to achieve a magnification of 1/3. Let's assume the new object distance is x cm.

Using the magnification formula:

magnification (m) = -v/u

1/3 = -(-20 cm)/(x cm)

1/3 = 20 cm/x cm

Cross-multiplying:

x cm = 20 cm * 3

x cm = 60 cm

Therefore, the object should be placed at a distance of 60 cm from the concave spherical mirror to achieve a magnification of 1/3.

**Conclusion**

By using the mirror formula and the magnification formula, we determined that when an object is placed at a distance of 40 cm from a concave spherical mirror and the magnification produced is 1/2, the object should be placed at a distance of 60 cm to achieve a magnification of 1/3.