Gravitational Potential Energy and Physics Fundamental Concepts

Scenario: While overtaking another cyclist, Ben increases his speed uniformly from $4.2\,ms^{-1}$ to $6.7\,ms^{-1}$ east over a time interval of $0.50\,s$ .
Tasks: * a) Calculate the magnitude of Ben's average acceleration during this time. * b) Determine how far Ben travels while overtaking. * c) Determine Ben's average speed during this time.

Define Gravity and gravitational field strength.
Identify the variables for calculating weight.
Define gravitational potential energy and explain where this energy comes from.
Identify the variables within the formula for finding gravitational potential energy.

Weight is defined as the gravitational force exerted on an object.
Calculation: Weight is calculated as a product of mass and gravity ( $W = m \times g$ ).
The average value of the gravitational field strength ( $g$ ) on Earth is $9.8\,m/s^{-2}$ .
Weight is dependent on the value of "g": * Scenario - Earth: If mass ( $m$ ) is $70\,kg$ and $g = 9.81\,N/kg$ , then $\text{Weight} = 70\,kg \times 9.81\,N/kg = 687\,N$ . * Scenario - Moon: If mass ( $m$ ) is $70\,kg$ and $g = 1.63\,N/kg$ , then $\text{Weight} = 70\,kg \times 1.63\,N/kg = 114\,N$ .

Weight: * Definition: The force that is generated by the earth's gravity on an object or mass. * Minimum value: Weight can be zero. * Quantity Type: Vector quantity. * Measurement Unit: Newton ( $N$ ). * Tool: Measured by a spring balance.
Mass: * Definition: The property of an object that tends to maintain its current state of rest or motion. * Minimum value: Mass cannot be zero. * Quantity Type: Scalar quantity. * Measurement Unit: Kilogram ( $kg$ ). * Tool: Ordinary balance is used to measure the mass.

1. What is the weight of a: * (a) $52\,kg$ mass? * (b) $2.6\,kg$ mass?
2. Determine the mass of bodies with the weights: * (a) $350\,N$ * (b) $5\,N$
3. Find the weight of a student of mass $48\,kg$ .
4. What is the weight of a $1000\,kg$ motor car?
5. What is the mass of a crate which weighs $1200\,N$ ?
6. A vehicle of mass $800\,kg$ has a horizontal force equal to its weight applied to it. Find the: * (a) weight of the vehicle * (b) acceleration produced.
7. A body of mass $4\,kg$ at rest is acted upon by a force equal to half its weight. Determine after $3\,s$ , the: * (a) velocity * (b) displacement.
8. A body is suspended on a spring balance and shows a weight of $20\,N$ . This same balance and body on the surface of the moon would show $3.2\,N$ . What is the acceleration due to gravity on the surface of the moon?

General Definition: The potential energy of an object is associated with its position relative to another object or within a field.
Forms of Potential Energy: * Chemical * Elastic * Gravitational * Magnetic * Nuclear * Spring

Definition: Gravitational potential is a measure of the amount of energy available to an object due to its position in a gravitational field.
Capacity for Change: Any object lifted above the Earth's surface has the capacity to cause change due to its position in the Earth's gravitational field.
Relationship to Work: The GPE of an object can be calculated from the constant amount of work that must be done against gravity to get the object into its position.
Theoretical Model (Weightlifter): If a weightlifter lifts a barbell at a constant speed, they must apply a lifting force equal to the force due to gravity ( $F_g$ ) over a displacement ( $\Delta h$ ).
Derivation: * $\text{Work} = \text{Force} \times \text{displacement}$ * $W = F_g \times \Delta h$ * Since $F_g = m \times g$ , then $W = m \times g \times \Delta h$ * Therefore, the change in GPE is $\Delta E_p = m \times g \times \Delta h$

Formula: $\Delta E_p = m \times g \times \Delta h$
Variables and Units: * $\Delta E_p$ : The change in gravitational potential energy, measured in Joules ( $J$ ). * $m$ : The mass of the object, measured in Kilograms ( $kg$ ). * $g$ : The magnitude of the gravitational field strength ( $9.8\,N\,kg^{-1}$ downwards on Earth). * $\Delta h$ : The change in height of the object, measured in Meters ( $m$ ).

When calculating potential energy, it is critical to define the height that corresponds to zero potential energy ( $E_p = 0$ ).
Terminology: This is known as the "reference level."
Standard reference levels are often the ground or sea level.

Example 1 (Barbell): * Scenario: A weightlifter lifts a barbell with a mass of $80.0\,kg$ from the floor to a height of $1.8\,m$ . * Calculation: $\Delta E_p = 80.0 \times 9.8 \times 1.8$
Example 2 (Weightlifter and Bar): * Scenario: A weightlifter ( $m = 60.0\,kg$ personally) lifts a $50.0\,kg$ bar through a distance of $45\,cm$ . * Calculation: Use $g = 9.8\,N\,kg^{-1}$ and convert height to meters ( $0.45\,m$ ).
Example 3 (Grocery Bag): * Scenario: A person lifts a $5.0\,kg$ grocery bag to a height of $30.0\,cm$ . * Working: $\Delta h = 0.30\,m$ . * Equation: $\Delta E_p = 5 \times 9.8 \times 0.30$ * Answer: $\Delta E_p = 15\,J$
Example 4 (Baby and Mattress): * Scenario: A father picks up a $6.0\,kg$ baby from a bed. The mattress is $70.0\,cm$ above ground. He holds the baby $125\,cm$ off the ground. * Working: $\Delta h = 125\,cm - 70.0\,cm = 55\,cm = 0.55\,m$ . * Equation: $\Delta E_p = 6 \times 9.80 \times 0.55$ * Answer: $\Delta E_p = 32\,J$

1. A rocket of mass $25\,kg$ reaches a height of $500\,m$ before falling. What is its maximum potential energy? * Solution: $125000\,J$
2. A drum of mass $80\,kg$ is rolled up a ramp onto a truck $1\,m$ above ground. What is its gain in potential energy? * Solution: $800\,J$
3. How much work is required to lift a bag of wheat of mass $50\,kg$ to a height of $2.4\,m$ ? * Solution: $1200\,J$
4. An object of mass $8\,kg$ loses $4000\,J$ of potential energy falling from a cliff. Determine the height of the cliff. * Solution: $50\,m$
5. What is the gain in potential energy of a pendulum bob of weight $15\,N$ raised vertically $0.1\,m$ ? * Solution: $1.5\,J$
6. A brick ( $2\,kg$ ) is dropped from a tower. If the striking velocity is $30\,m\,s^{-1}$ , how high is the tower? * Solution: $45\,m$
7. How high does a ball ( $0.5\,kg$ ) travel if given upward kinetic energy of $46\,J$ ? * Solution: $9.2\,m$
8. Find the mass of a stone which, when dropped $50\,m$ , strikes the bottom with $200\,J$ of kinetic energy. * Solution: $0.4\,kg$
9. A pendulum bob ( $0.5\,kg$ ) swings through its lowest point at $0.8\,m\,s^{-1}$ . What is the maximum vertical height it travels? * Solution: $0.032\,m$

An arrow ( $0.1\,kg$ ) is fired upwards at $30\,m\,s^{-1}$ . When its velocity decreases to $10\,m\,s^{-1}$ , determine: * (a) Kinetic energy (Solution: $5\,J$ ) * (b) Potential energy (Solution: $40\,J$ )