A 350-g air track cart on a horizontal air track is attached to a string that goes over a pulley with a moment of
inertia of 6.00 × 10-6 kg·m2 and a radius of 1.35 cm. The string is pulled vertically downward by a force of 2.50
N. What is the acceleration of the cart?
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To find the acceleration of the cart, we need to consider the forces acting on it. In this case, we have the force due to gravity and the tension in the string.
1. First, let's calculate the force due to gravity acting on the cart. The force due to gravity can be calculated using the formula:
F_gravity = m * g
where m is the mass of the cart (350 g = 0.35 kg) and g is the acceleration due to gravity (9.8 m/s^2).
Plugging in the values, we get:
F_gravity = 0.35 kg * 9.8 m/s^2
2. Next, let's calculate the torque due to the force applied by the string. Torque is calculated using the formula:
Torque = F * r
where F is the force applied by the string (2.50 N) and r is the radius of the pulley (1.35 cm = 0.0135 m).
Plugging in the values, we get:
Torque = 2.50 N * 0.0135 m
3. Now, let's calculate the torque due to the force of gravity acting on the pulley. The torque due to gravity can be calculated using the formula:
Torque_gravity = m * g * r
where m is the mass of the pulley (6.00 × 10^-6 kg) and g is the acceleration due to gravity (9.8 m/s^2), and r is the radius of the pulley (1.35 cm = 0.0135 m).
Plugging in the values, we get:
Torque_gravity = 6.00 × 10^-6 kg * 9.8 m/s^2 * 0.0135 m
4. The net torque acting on the pulley is the difference between the torque due to the force applied by the string and the torque due to gravity. Net torque can be calculated using the formula:
Net Torque = Torque - Torque_gravity
Plugging in the values, we get:
Net Torque = (2.50 N * 0.0135 m) - (6.00 × 10^-6 kg * 9.8 m/s^2 * 0.0135 m)
5. The net torque is equal to the moment of inertia of the pulley (I) multiplied by the angular acceleration (α). Net Torque can be calculated using the formula:
Net Torque = I * α
Rearranging the formula, we can find the angular acceleration:
α = Net Torque / I
Plugging in the values, we get:
α = Net Torque / (6.00 × 10^-6 kg·m^2)
6. Finally, the linear acceleration of the cart can be calculated using the formula:
a = α * r
where α is the angular acceleration we found in the previous step and r is the radius of the pulley (0.0135 m).
Plugging in the values, we get:
a = α * 0.0135 m
These calculations will give us the acceleration of the cart.