A cart with a mass of 5.35 kg is placed on a frictionless ramp at an angle of 14.2. The cart is released and rolls down the ramp. Find the acceleration.
a = g sinθ
To find the acceleration of the cart rolling down the ramp, we can use the principles of physics. The acceleration can be determined using the equation:
acceleration = gravitational force * sin(angle) / mass
In this equation:
- gravitational force refers to the force due to gravity, which is the product of the mass of the cart (5.35 kg) and the acceleration due to gravity (9.8 m/s^2)
- angle refers to the angle of the ramp (14.2 degrees)
- mass refers to the mass of the cart (5.35 kg)
Substituting the values into the equation, we get:
acceleration = (5.35 kg * 9.8 m/s^2) * sin(14.2 degrees) / 5.35 kg
Simplifying the equation, the mass of the cart cancels out:
acceleration = 9.8 m/s^2 * sin(14.2 degrees)
Now, we can calculate the acceleration:
acceleration ≈ 2.486 m/s^2
Therefore, the acceleration of the cart rolling down the ramp is approximately 2.486 m/s^2.
To find the acceleration of the cart as it rolls down the ramp, we can use the equation for gravitational force along the ramp:
F_gravity = m * g * sin(θ),
where
F_gravity is the gravitational force along the ramp,
m is the mass of the cart (5.35 kg in this case),
g is the acceleration due to gravity (approximately 9.8 m/s^2), and
θ is the angle of the ramp (14.2 degrees in this case).
First, we need to convert the angle from degrees to radians. We can use the formula:
θ_radians = θ_degrees * π / 180.
θ_radians = 14.2 * π/180
θ_radians ≈ 0.248.
Now, we can substitute the values into the equation to find the gravitational force along the ramp:
F_gravity = 5.35 kg * 9.8 m/s^2 * sin(0.248).
F_gravity ≈ 5.35 kg * 9.8 m/s^2 * 0.2468.
F_gravity ≈ 12.7461 N.
Next, we need to calculate the net force acting on the cart. Since there is no friction, the net force is equal to the gravitational force along the ramp:
F_net = F_gravity.
F_net = 12.7461 N.
Finally, we can use Newton's second law of motion, F = m * a, to find the acceleration:
F_net = m * a.
12.7461 N = 5.35 kg * a.
Solving for a, we find:
a ≈ 2.385 m/s^2.
Therefore, the acceleration of the cart rolling down the ramp is approximately 2.385 m/s^2.