a boy of mass 40kg is sliding down the spiral slide at a constant speed such that his position is measured by
theta=(0.7t) rad
and
z=(-0.5t) m
where t is in seconds. How do you find the components of forces Fr, Ftheta and Fz which the slide exerts on the boy at the instant when t=2s. Neglect the size of the boy.
I need to know r
w = d theta/dt = .7
Fr = 40 r w^2 = 40 r (.49) inward (negative r direction)
the tan of angle of the slide down from horizontal = .5/(wr) = .5/(.7r) = .714/r
Fz = weight of boy up = 40*9.8 = 392 N
Ftheta = weight tan angle = 392 (.714/r)
To find the components of forces exerted by the slide on the boy at the instant when t = 2s, we'll use the given equations for position:
θ = (0.7t) rad (Equation 1)
z = (-0.5t) m (Equation 2)
First, let's differentiate the position equations with respect to time to find the velocity components:
dθ/dt = 0.7 rad/s (Equation 3)
dz/dt = -0.5 m/s (Equation 4)
Next, we'll differentiate the velocity equations with respect to time to find the acceleration components:
d²θ/dt² = 0 rad/s² (Equation 5)
d²z/dt² = 0 m/s² (Equation 6)
Since the boy is sliding down the slide at a constant speed, the acceleration components are both zero, which means there is no radial (Fr) or vertical (Fz) force acting on the boy. The only force acting on the boy is in the tangential direction (Fθ).
Now, let's use Newton's second law to find the force in the tangential direction:
Fθ = m * aθ,
where m is the mass of the boy and aθ is the tangential acceleration. Since aθ = d²θ/dt² = 0, the tangential force is also zero:
Fθ = 0.
Therefore, the components of forces exerted by the slide on the boy at t = 2s are as follows:
Fr = 0 N
Fθ = 0 N
Fz = 0 N
To find the components of forces Fr, Ftheta, and Fz at the instant when t = 2s, we need to calculate the derivatives of theta and z with respect to time, and use them in the force equations.
Given:
theta = 0.7t (in radians)
z = -0.5t (in meters)
We will find the velocity components first, and then use them to find the forces.
Step 1: Calculate the derivatives of theta and z.
d(theta)/dt = d(0.7t)/dt = 0.7 (rad/s)
dz/dt = d(-0.5t)/dt = -0.5 (m/s)
Step 2: Calculate the velocity components.
v_r = dθ/dt = 0.7 (rad/s)
v_theta = r * dθ/dt = r * 0.7 (rad/s), where r is the radius of the spiral slide
v_z = dz/dt = -0.5 (m/s)
Step 3: Calculate the forces.
Fr = m * (v_theta)^2 / r
Ftheta = -m * v_r * v_theta / r
Fz = m * v_z
Plugging in the values:
m = 40 kg
v_theta = r * 0.7 (rad/s)
v_r = 0.7 (rad/s)
v_z = -0.5 (m/s)
Fr = (40 kg) * (0.7 rad/s)^2 / r
Ftheta = -(40 kg) * (0.7 rad/s) * (0.7 rad/s) / r
Fz = (40 kg) * (-0.5 m/s)
Since the size of the boy is neglected, the forces are applied at a single point, hence they are not dependent on the position of the boy on the slide.
Please note that the values of r (radius of the spiral slide) are not provided in the question. You would need to know the radius of the slide to compute the values of Fr and Ftheta accurately.