A dynamics cart with a friction pad is placed at the top of an inclined track and released from rest. The cart accelerates down the incline at a rate of .60m/s^2. If the track is angled at 10 degrees above the horizontal, determine the coefficient of kinetic friction between the cart and the track.
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To determine the coefficient of kinetic friction between the cart and the track, we can use the following steps:
Step 1: Identify the known values:
- Acceleration of the cart down the incline (a) = 0.60 m/s^2
- Angle of the incline (θ) = 10 degrees
Step 2: Calculate the gravitational force acting on the cart:
The force due to gravity (Fg) can be calculated using the equation: Fg = m * g, where:
- m is the mass of the cart
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Step 3: Calculate the component of the gravitational force parallel to the incline:
The component of the gravitational force acting parallel to the incline can be calculated using the equation: Fpar = Fg * sin(θ)
Step 4: Calculate the net force acting on the cart:
The net force (Fnet) can be calculated using the equation: Fnet = m * a, where:
- m is the mass of the cart
- a is the acceleration of the cart down the incline
Step 5: Determine the force of friction:
The force of friction (Ffr) can be calculated using the equation: Ffr = Fnet - Fpar
Step 6: Calculate the normal force:
The normal force (Fn) is the force exerted by the track on the cart perpendicular to the incline. It can be calculated using the equation: Fn = Fg * cos(θ)
Step 7: Determine the coefficient of kinetic friction:
The coefficient of kinetic friction (μk) can be calculated using the equation: μk = Ffr / Fn
So, by following these steps, you can determine the coefficient of kinetic friction between the cart and the track.
To determine the coefficient of kinetic friction between the cart and the track, we can use the following steps:
Step 1: Determine the acceleration along the direction of the incline.
Given: acceleration (a) = 0.60 m/s^2
Step 2: Convert the angle of the incline from degrees to radians.
Given: angle (θ) = 10 degrees
θ (in radians) = θ (in degrees) × π / 180
θ (in radians) = 10 × π / 180 = 0.1745 radians (rounded to four decimal places)
Step 3: Calculate the component of the gravitational force acting along the incline.
The component of the gravitational force parallel to the incline is given by:
F_parallel = m × g × sin(θ)
where m is the mass of the cart and g is the acceleration due to gravity (9.8 m/s^2).
Since the cart is not moving vertically, the normal force and the gravitational force cancel each other out, so F_normal = m × g × cos(θ)
Given: m = mass of the cart
Step 4: Calculate the net force along the incline.
The net force acting on the cart is given by:
F_net = F_parallel - F_friction
where F_friction is the frictional force opposing the motion.
Step 5: Calculate the frictional force.
The frictional force is given by the equation:
F_friction = μ × F_normal
where μ is the coefficient of kinetic friction.
Step 6: Substitute the values into the equations.
Using the values and equations from previous steps, we can rewrite the net force equation as follows:
F_net = m × a
F_net = m × g × sin(θ) - μ × m × g × cos(θ)
m × a = m × g × sin(θ) - μ × m × g × cos(θ)
Divide both sides by m × g:
a = g × sin(θ) - μ × g × cos(θ)
Step 7: Solve the equation for the coefficient of kinetic friction (μ).
μ = (g × sin(θ) - a) / (g × cos(θ))
Step 8: Substitute the known values into the equation and calculate.
Substitute the known values:
g = 9.8 m/s^2
θ (in radians) = 0.1745 radians
a = 0.60 m/s^2
μ = (9.8 × sin(0.1745) - 0.60) / (9.8 × cos(0.1745))
μ ≈ 0.100 (rounded to three decimal places)
Therefore, the coefficient of kinetic friction between the cart and the track is approximately 0.100.