A. The gravitational potential energy (PE) of the cart can be calculated using the equation:
PE = M * g * h
Where:
- M is the mass of the cart
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height of the cart above the reference point (in this case, the bottom of the ramp)
Since the ramp is inclined, we need to calculate the height h in terms of the distance along the ramp (X) and the angle of the ramp (θ).
Considering the given angle θ and the distance X along the ramp, we can find the vertical height (h) using trigonometry. The vertical height (h) can be expressed as:
h = X * sin(θ)
Substituting this value into the gravitational potential energy equation, we get:
PE = M * g * (X * sin(θ))
B. As the cart rolls down the inclined ramp, the kinetic energy (KE), gravitational potential energy (PE), and total mechanical energy (TME) change.
- Kinetic Energy (KE):
The kinetic energy of the cart increases as it rolls down the ramp. This is because the potential energy of the cart is being converted into kinetic energy due to the cart's motion. The equation for the kinetic energy of the cart is:
KE = 1/2 * M * v^2
Where:
- M is the mass of the cart
- v is the velocity of the cart
As the cart accelerates down the ramp, its velocity increases, resulting in an increase in kinetic energy.
- Gravitational Potential Energy (PE):
The gravitational potential energy of the cart decreases as it rolls down the ramp. This is because the cart is moving closer to the reference point with zero potential energy (the bottom of the ramp). The equation for potential energy is:
PE = M * g * h
Since the vertical height (h) above the reference point decreases as the cart moves down the ramp, the gravitational potential energy decreases.
- Total Mechanical Energy (TME):
The total mechanical energy of the cart remains constant (ignoring factors like friction) throughout its motion down the ramp. The equation for total mechanical energy is the sum of kinetic energy and gravitational potential energy:
TME = KE + PE
Since the kinetic energy increases and the gravitational potential energy decreases, the sum of the two remains constant, resulting in a constant total mechanical energy.