The ratio of masses of two planets is 2:3 and the ratio of their radii is 4:7 Find the ratio of their accelerations due to gravity. - Science

Cosmic Calculations: How Strong is Gravity on Other Worlds?

We all feel gravity's pull every day, but have you ever wondered how it might differ on other planets? The "weight" we feel is determined by a planet's acceleration due to gravity, a value that depends on both its mass and its size.

A more massive planet will have a stronger gravitational pull, but a larger radius will weaken that pull at the surface. Understanding this balance allows us to compare different worlds, even without visiting them. Let's explore this with a fascinating problem.

The Problem

The ratio of the masses of two planets is 2:3 and the ratio of their radii is 4:7.

Question: Find the ratio of their accelerations due to gravity.

The Science Behind the Solution

The acceleration due to gravity, denoted by '$g$', is calculated using Newton's law of universal gravitation. The formula is:

$$g = \frac{GM}{R^2}$$

Here, '$G$' is the gravitational constant, '$M$' is the mass of the planet, and '$R$' is its radius. Since we are looking for the ratio of the gravities of two planets (let's call them Planet 1 and Planet 2), we can set up the following relation:

$$\frac{g_1}{g_2} = \frac{GM_1/R_1^2}{GM_2/R_2^2}$$

The gravitational constant '$G$' cancels out, simplifying the equation to:

$$\frac{g_1}{g_2} = \frac{M_1}{M_2} \times \left(\frac{R_2}{R_1}\right)^2$$

Solving the Problem Step-by-Step

Let's list what we know from the problem:

• Ratio of masses ($M_1 : M_2$): 2 : 3, which means $\frac{M_1}{M_2} = \frac{2}{3}$
• Ratio of radii ($R_1 : R_2$): 4 : 7, which means $\frac{R_1}{R_2} = \frac{4}{7}$

Notice that our formula needs the ratio $\frac{R_2}{R_1}$. We can easily find this by inverting the ratio we have: if $\frac{R_1}{R_2} = \frac{4}{7}$, then $\frac{R_2}{R_1} = \frac{7}{4}$.

Now, we plug these ratios into our simplified formula:

$$\frac{g_1}{g_2} = \frac{2}{3} \times \left(\frac{7}{4}\right)^2$$

First, we square the radii ratio:

$$\frac{g_1}{g_2} = \frac{2}{3} \times \frac{49}{16}$$

Finally, we multiply the fractions:

$$\frac{g_1}{g_2} = \frac{2 \times 49}{3 \times 16} = \frac{98}{48}$$

This fraction can be simplified by dividing the numerator and denominator by 2:

$$\frac{g_1}{g_2} = \frac{49}{24}$$

Answer: The ratio of the accelerations due to gravity ($g_1 : g_2$) is 49:24.

The Takeaway

This result shows how both mass and radius play a crucial role. Even though Planet 2 is more massive (3/2 times the mass of Planet 1), its much larger radius means its surface gravity is actually weaker. It's a perfect example of how cosmic properties are about more than just one factor!

A mechanic unscrew a nut by applying a force of 140 N with a spanner of length 40 cm. What should be the length of the spanner if a force of 40 N is applied to unscrew the same nut? - Science

The Physics of a Spanner: Understanding Torque

Ever wondered why it's easier to loosen a stubborn nut with a longer wrench? The answer isn't magic; it's physics! Specifically, it's a fundamental concept called torque, or the moment of force.

Torque is the measure of the force that can cause an object to rotate around an axis. Think of it as a "turning" or "twisting" force. It’s at play in many of our daily activities, from pushing open a door (you push farthest from the hinges, right?) to pedaling a bicycle.

To see this principle in action, let's break down a classic physics problem.

The Problem

A mechanic unscrews a nut by applying a force of 140 Newtons (N) with a spanner that is 40 centimeters (cm) long.

Question: What should be the length of the spanner if a force of only 40 N is applied to unscrew the exact same nut?

The Science Behind the Solution

To loosen the nut, a specific amount of torque is required. This required torque is a constant. It doesn't change whether you use a short spanner with a lot of force or a long spanner with less force. The turning effect must be the same.

The formula for torque is simple:

$$Torque = Force \times \text{Perpendicular Distance from the pivot}$$

In our case, the "distance" is the length of the spanner. Let's call our first scenario (Case 1) and our second scenario (Case 2). The core principle is:

$$Torque_1 = Torque_2$$

Which means:

$$(Force_1 \times \text{Length}_1) = (Force_2 \times \text{Length}_2)$$

Solving the Problem Step-by-Step

Let's list what we know:

• Force 1 ($F_1$): 140 N
• Length 1 ($d_1$): 40 cm
• Force 2 ($F_2$): 40 N
• Length 2 ($d_2$): ? (This is what we need to find)

Now, we plug these values into our equation:

$$F_1 \times d_1 = F_2 \times d_2$$

$$140 \text{ N} \times 40 \text{ cm} = 40 \text{ N} \times d_2$$

To solve for $d_2$, we can rearrange the equation:

$$d_2 = \frac{(140 \text{ N} \times 40 \text{ cm})}{40 \text{ N}}$$

As you can see, the "40 N" on the top and bottom of the fraction cancel each other out.

$$d_2 = 140 \text{ cm}$$

Answer: The mechanic would need a spanner that is 140 cm long.

The Takeaway

This problem beautifully illustrates the inverse relationship between force and the length of the lever arm when torque is constant.

• Less force? You need a longer lever.
• Shorter lever? You need to apply more force.

So, the next time you're struggling with a tight bolt, remember your physics lesson and grab a longer wrench!