Question

If the mass of the bob is \(4 \mathrm{~kg}\) and the maximum height it reaches is \(1.8 \mathrm{~m}\), what is the speed of the bob as it swings through the lowest position? (A) \(1 \mathrm{~m} / \mathrm{s}\) (B) \(3 \mathrm{~m} / \mathrm{s}\) (C) \(5 \mathrm{~m} / \mathrm{s}\) (D) \(6 \mathrm{~m} / \mathrm{s}\)

Short answer

The velocity of the bob at the lowest position is approximately 2.97 m/s, which is closest to option (B), 3 m/s.

Step-by-step solution

Determine the potential energy at the maximum height

Using the given information, first, we will find the gravitational potential energy at the maximum height, which can be calculated using the following formula: \( U = mgh \) Where \(m\) = mass of the bob (4 kg) \(g\) = acceleration due to gravity (approximately 9.81 m/s²) \(h\) = maximum height (1.8 m) Substituting the values: \( U = 4 kg × 9.81 \frac{m}{s²} × 1.8 m \) \( U = 70.584 \mathrm{~J} \) (joules)

Calculate the kinetic energy at the lowest position

Since the mechanical energy is conserved, the potential energy at the highest position must be equal to the kinetic energy when the bob passes through its lowest position. So, the kinetic energy at the lowest point is 70.584 J.

Determine the velocity at the lowest position

The formula for kinetic energy is: \( K = \frac{1}{2}mv² \) We can now use the given mass and the kinetic energy calculated in step 2 to find the velocity, v: \(70.584 J = \frac{1}{2}(4 kg)v²\) To solve for the speed, v, we first divide both sides by 2: \(35.292 J = (4 kg)v²\) Now, divide by the mass (4 kg): \(8.823 = v²\) Finally, take the square root of both sides: \(v = \sqrt{8.823} \approx 2.97 \frac{m}{s}\) The velocity of the bob at the lowest position is approximately 2.97 m/s, which is closest to option (B), 3 m/s.

Key concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. In simple terms, it is the energy stored when an object is lifted to a certain height. The higher an object is lifted, the more gravitational potential energy it has. The formula to calculate gravitational potential energy is given by:

• \( U = mgh \)
• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity, which is approximately \(9.81 \, \text{m/s}^2\) on Earth
• \( h \) is the height above a reference point

For example, if you have a bob with a mass of 4 kilograms raised to a height of 1.8 meters, the gravitational potential energy would be:- \( U = 4 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 1.8 \, \text{m} = 70.584 \, \text{J} \)This energy represents the potential for the bob to do work as it moves downward, converting this potential energy into kinetic energy.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. Any object in motion has kinetic energy. This energy depends on two factors: the mass of the object and its velocity. The relationship is expressed through the formula:

• \( K = \frac{1}{2}mv^2 \)
• \( m \) is the mass of the object
• \( v \) is the velocity of the object

Imagine the bob reaching its lowest position in its swing, the potential energy it had at the top is converted entirely into kinetic energy, as there is no height to provide potential energy. Using the calculated potential energy (70.584 J), this is also the kinetic energy when the bob is at its lowest position. You can find the speed using:- \( 70.584 = \frac{1}{2}(4)v^2 \)
- \( v^2 = 35.292 \)
- \( v = \sqrt{8.823} \approx 2.97 \, \text{m/s} \)
Thus, the speed of the bob at its lowest point is about 3 m/s, demonstrating how potential energy is transformed into kinetic energy.

Conservation of Energy

The conservation of energy principle is a fundamental concept in physics which states that energy in a closed system remains constant. Energy cannot be created or destroyed; it can only be transformed from one form to another. In our scenario, this principle shows how gravitational potential energy converts to kinetic energy as the bob swings down to its lowest position.- At the maximum height, the bob has maximum gravitational potential energy and no kinetic energy since it momentarily stops.- As it swings down, potential energy decreases and kinetic energy increases.- At the lowest position, all potential energy is converted to kinetic energy, resulting in maximum speed.Through these transformations, the total mechanical energy (sum of potential and kinetic energy) remains the same:\[U_{\text{initial}} + K_{\text{initial}} = U_{\text{final}} + K_{\text{final}}\]Using this principle effectively allows us to find quantities like speed when specified energies are provided or assumed. Here, by using conservation of energy, we arrived at the bob's speed as 3 m/s without needing to know the time taken or forces involved directly.