A 0.55-kg air cart is attached to a spring and allowed to oscillate.
If the displacement of the air cart from equilibrium is x=(10.0 cm)cos[(2.00 s ^-1)t + pi], find the maximum kinetic energy of the cart.
Find the maximum force exerted on it by the spring.
To find the maximum kinetic energy of the cart, we can use the equation for kinetic energy, given by K = (1/2)mv^2, where K is the kinetic energy, m is the mass of the cart, and v is the velocity of the cart.
The velocity of the cart can be found by taking the derivative of the displacement equation with respect to time. Let's calculate it step by step.
Given:
Mass of the cart, m = 0.55 kg
Displacement from equilibrium, x = (10.0 cm)cos[(2.00 s ^-1)t + pi] = (10.0 cm)cos(2πt + π)
1. Find the velocity v:
The derivative of x with respect to time gives us the velocity.
dx/dt = v = -10.0 cm * (2.00 s ^-1) * sin[(2.00 s ^-1)t + pi]
Now we need to convert cm/s to m/s.
1 m = 100 cm
1 s = 1 s
v = -(10.0 cm/m) * (2.00 s ^-1) * sin[(2.00 s ^-1)t + π]
v = -0.2 m/s * sin[(2.00 s ^-1)t + π]
2. Find the maximum velocity:
The maximum velocity occurs when sin[(2.00 s ^-1)t + π] = 1 (maximum value for sin function).
So, v_max = -0.2 m/s * 1 = -0.2 m/s (negative sign because the direction of velocity depends on the displacement equation).
3. Find the maximum kinetic energy:
K_max = (1/2) * m * v_max^2
K_max = (1/2) * 0.55 kg * (-0.2 m/s)^2
Now we can calculate K_max.
K_max = (1/2) * 0.55 kg * 0.04 m^2/s^2
K_max = 0.011 J
Therefore, the maximum kinetic energy of the cart is 0.011 J.
Now for the maximum force exerted by the spring:
The force exerted by the spring is given by Hooke's law, F = -kx, where F is the force, k is the spring constant, and x is the displacement.
The maximum force occurs when the displacement is maximum, so we can use the maximum displacement value from the given equation: x_max = 10.0 cm = 0.1 m.
4. Find the maximum force exerted by the spring:
F_max = -k * x_max
To find the spring constant, we need more information. The spring constant can be determined experimentally or given in the problem statement.
Please provide the value for the spring constant (k) to proceed with finding the maximum force exerted by the spring.