Question

Consider the two spheres shown here, one made of silver and the other of aluminum.(a) What is the mass of each sphere in \(\mathrm{kg}\) ( b) The force of gravity acting on an object is \(F=m g\) where \(m\) is the mass of an object and \(g\) is the acceleration of gravity \(\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right) .\) How much work do you do on each sphere it you raise it from the floor to a height of 2.2 \(\mathrm{m} ?(\mathrm{c})\) Does the act of lifting the sphere off the ground increase the potential energy of the aluminum sphere by a larger, smaller, or same amount as the silver sphere? (d) If you release the spheres simultaneously, they will have the same velocity when they hit the ground. Will they have the same kinetic energy? If not, which sphere will have more kinetic energy? \([\) Section 1.4\(]\)

Short answer

The mass of the silver sphere is approximately 0.148 kg and the mass of the aluminum sphere is approximately 0.038 kg. The work done on the silver sphere is approximately 3.207 J, while the work done on the aluminum sphere is approximately 0.822 J. The potential energy of the aluminum sphere increases by a smaller amount than the potential energy of the silver sphere. When released simultaneously, the spheres will have different kinetic energies, with the silver sphere having more kinetic energy than the aluminum sphere.

Step-by-step solution

(a) Calculating the Mass of Each Sphere

First, we need to find the volume and mass of each sphere. Given densities of silver (\(\rho_{Ag}\) = 10,500 kg/m³) and aluminum (\(\rho_{Al}\) = 2,700 kg/m³), and radius of both spheres (r = 1.5 cm = 0.015 m). Volume of a sphere can be calculated as follows: \(V = \frac{4}{3} \pi r^3\) Calculate the volume of each sphere: \(V = \frac{4}{3} \pi (0.015)^3 = 1.413 \times 10^{-5} m^3\) Now, to calculate the mass of each sphere: \(m_{Ag} = \rho_{Ag} * V_{Ag} = 10,500 kg/m³ * 1.413 \times 10^{-5} m^3 = m_{Ag} \approx 0.148 kg\) \(m_{Al} = \rho_{Al} * V_{Al} = 2,700 kg/m³ * 1.413 \times 10^{-5} m^3 = m_{Al} \approx 0.038 kg\)

(b) Calculating Work Done on Each Sphere

Next, we need to calculate the work done when each sphere is raised to a height of 2.2 m. We will use the formula: \(W = mgh\) Where: \(m\) - mass of the sphere \(g\) - acceleration due to gravity (9.8 m/s²) \(h\) - height (2.2 m) Calculate the work done on each sphere: \(W_{Ag} = m_{Ag}gh = 0.148 kg * 9.8 m/s² * 2.2 m \approx 3.207 J\) \(W_{Al} = m_{Al}gh = 0.038 kg * 9.8 m/s² * 2.2 m \approx 0.822 J\)

(c) Comparing Potential Energy Increase

Here, we need to determine if the aluminum sphere has a larger, smaller, or the same potential energy increase as the silver sphere when lifted off the ground. Since the work done is equal to the change in potential energy, we can compare the work done in part (b). Since \(W_{Ag} > W_{Al}\), the potential energy of the aluminum sphere increases by a smaller amount than the potential energy of the silver sphere.

(d) Comparing Kinetic Energy

Now, we will discuss whether both spheres - when released simultaneously - have the same kinetic energy or not before they hit the ground. Both spheres have the same height, and in the absence of air resistance, they will have the same velocity when they hit the ground. However, their masses are different; since kinetic energy depends upon mass and velocity, they will not have the same kinetic energy when they hit the ground. The formula for kinetic energy is: \(K = \frac{1}{2}mv^2\) As \(m_{Ag} > m_{Al}\), the silver sphere will have more kinetic energy than the aluminum sphere before hitting the ground.

Key concepts

Mass Calculation

Mass calculation is vital in physics as it helps us understand the amount of matter in an object. To determine the mass of an object like a sphere, we first need to calculate its volume. For a sphere, the volume (V) is given by the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius. Knowing the volume allows us to find the mass because mass is the product of volume and density.

• The density (\rho) of a material is its mass per unit volume, typically expressed in \( kg/m^3 \).
• To find mass (\m), use the expression \( m = \rho \times V \).

For instance, if you have a silver sphere with a density (\rho_{Ag}) of 10,500 \( kg/m³ \) and an aluminum sphere with a density (\rho_{Al}) of 2,700 \( kg/m³ \), you can substitute these values into the formula with the respective volumes to calculate their masses.

Gravitational Force

Gravitational force is the attractive force that objects exert on each other due to their masses. This force is crucial for understanding how objects interact in a field created by massive bodies like Earth. Newton's law of universal gravitation tells us that the gravitational force (\F) on an object is calculated using the equation \( F = mg \).

• Here, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity.
• On Earth, the standard gravitational acceleration (g) is approximately 9.8 \( m/s^2 \).

Thus, for each sphere, multiplying its calculated mass by the gravitational acceleration allows us to find the gravitational force acting on it. This principle explains why objects of different masses fall at the same rate, assuming no air resistance.

Potential Energy

Potential energy is a form of energy that is stored in an object due to its position in a force field, such as gravity. When lifting an object to a height, we increase its gravitational potential energy. This energy is calculated using the formula \( PE = mgh \), where:

• \( m \) is the mass,
• \( g \) is the gravitational acceleration, and
• \( h \) is the height raised.

The work done on lifting an object against gravity equates to the increase in its potential energy. For example, raising two spheres of different masses to the same height means they will have different potential energy increases. As calculated earlier, the silver sphere has more potential energy increase than the aluminum sphere because it has more mass. Greater mass leads to greater potential energy at a given height.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. Once a raised object is dropped, its potential energy converts to kinetic energy as it speeds toward the ground. The formula to calculate kinetic energy (\K) is \( K = \frac{1}{2}mv^2 \):

• \( m \) is the mass of the object,
• \( v \) is its velocity.

Even though spheres dropped from the same height will reach the ground at the same velocity (ignoring air resistance), their kinetic energies will differ because kinetic energy is also dependent on mass. The heavier silver sphere will have more kinetic energy upon impact compared to the lighter aluminum sphere, due to its greater mass converting more gravitational potential energy into kinetic form before hitting the ground.