Answer :
To determine which change to an object would double its potential energy, we need to understand the formula for gravitational potential energy, which is given by:
[tex]\[ \text{Potential Energy (PE)} = m \times g \times h \][/tex]
where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground.
Now let's examine each of the given options:
A. Increasing its height to twice its original value
- If we increase the height to twice its original value, the new height is \( 2h \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = m \times g \times (2h) \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = 2 \times (m \times g \times h) = 2 \text{PE} \][/tex]
- This means that the potential energy would be doubled.
B. Reducing its mass to one-half of its original value
- If we reduce the mass to one-half its original value, the new mass is \( \frac{m}{2} \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = \left(\frac{m}{2}\right) \times g \times h \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = \frac{1}{2} \times (m \times g \times h) = \frac{1}{2} \text{PE} \][/tex]
- This means that the potential energy would be halved, not doubled.
C. Increasing its mass to four times its original value
- If we increase the mass to four times its original value, the new mass is \( 4m \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = (4m) \times g \times h \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = 4 \times (m \times g \times h) = 4 \text{PE} \][/tex]
- This means that the potential energy would be quadrupled, not doubled.
D. Reducing its height to one-half of its original value
- If we reduce the height to one-half its original value, the new height is \( \frac{h}{2} \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = m \times g \times \left(\frac{h}{2}\right) \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = \frac{1}{2} \times (m \times g \times h) = \frac{1}{2} \text{PE} \][/tex]
- This means that the potential energy would be halved, not doubled.
From these calculations, we can see that the correct choice is:
A. Increasing its height to twice its original value
This change will double the potential energy of the object.
[tex]\[ \text{Potential Energy (PE)} = m \times g \times h \][/tex]
where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground.
Now let's examine each of the given options:
A. Increasing its height to twice its original value
- If we increase the height to twice its original value, the new height is \( 2h \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = m \times g \times (2h) \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = 2 \times (m \times g \times h) = 2 \text{PE} \][/tex]
- This means that the potential energy would be doubled.
B. Reducing its mass to one-half of its original value
- If we reduce the mass to one-half its original value, the new mass is \( \frac{m}{2} \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = \left(\frac{m}{2}\right) \times g \times h \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = \frac{1}{2} \times (m \times g \times h) = \frac{1}{2} \text{PE} \][/tex]
- This means that the potential energy would be halved, not doubled.
C. Increasing its mass to four times its original value
- If we increase the mass to four times its original value, the new mass is \( 4m \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = (4m) \times g \times h \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = 4 \times (m \times g \times h) = 4 \text{PE} \][/tex]
- This means that the potential energy would be quadrupled, not doubled.
D. Reducing its height to one-half of its original value
- If we reduce the height to one-half its original value, the new height is \( \frac{h}{2} \).
- The new potential energy would be: [tex]\[ \text{PE}_{\text{new}} = m \times g \times \left(\frac{h}{2}\right) \][/tex]
- Simplifying this, we get: [tex]\[ \text{PE}_{\text{new}} = \frac{1}{2} \times (m \times g \times h) = \frac{1}{2} \text{PE} \][/tex]
- This means that the potential energy would be halved, not doubled.
From these calculations, we can see that the correct choice is:
A. Increasing its height to twice its original value
This change will double the potential energy of the object.
