MD Robotics is the company that built the Canadarm for the International Space Station. Engineers (including
a friend of Mr. Meyer) are working on a way to use the arm to launch small satellites. The idea is to attach a
launch pad (1200 kg) to the hand (350 kg) of the arm. A small satellite (300 kg) is mounted on the launch pad.
A coiled spring (k = 1.7 x 10 5 N/m) is compressed and when it is released, it pushes the satellite off the launch
pad with a speed of 2 m/s as measured by the international space station.
You are part of the team helping to design this system. A junior engineer presents you with a rough design and
the specifications listed above. You want to: (1) check how far the spring should be compressed. You know
that this spring loses 9.7% of its stored energy to thermal energy.
Due to the launch, the hand and launch pad will recoil. After the launch, the motor of the arm will exert a force
to slow down the hand plus launch pad. The hand should stop after a 20 cm distance. You want to: (2) check
that the force required will not exceed the 2.4 x10 4 N that the motor is designed for.
i couldn't figure it out either tell me if you got any responses
(1) To check how far the spring should be compressed, we need to determine the amount of stored energy required to launch the satellite.
Given:
Mass of the launch pad (m1) = 1200 kg
Mass of the hand (m2) = 350 kg
Mass of the satellite (m3) = 300 kg
Speed of satellite (v) = 2 m/s
The total mass of the system is:
m_total = m1 + m2 + m3
To calculate the stored energy in the compressed spring, we can use the formula:
E = (1/2) k x^2
Where E is the energy, k is the spring constant, and x is the displacement of the spring.
Let's assume x as the distance the spring needs to be compressed.
The energy required to launch the satellite is given by:
E_required = (1 - 0.097) x E
The energy E can be calculated as:
E = (1/2) k x^2
Now, we can substitute the known values and solve for x:
E_required = (1 - 0.097) x (1/2) k x^2
Simplifying the equation, we have:
E_required = 0.903 x (1/2) k x^2
Since the speed of the satellite is given as 2 m/s, the kinetic energy can be equated to the stored energy:
E_required = (1/2) m3 v^2
Substituting the known values, we have:
0.903 x (1/2) k x^2 = (1/2) m3 v^2
Simplifying further:
0.903 x k x^2 = m3 v^2
Now we can solve for x:
x^2 = (m3 v^2) / (0.903 x k)
x = √[(m3 v^2) / (0.903 x k)]
Substituting the given values:
x = √[(300 kg x (2 m/s)^2) / (0.903 x (1.7 x 10^5 N/m))]
x = √[(300 x 4) / (0.903 x 1.7 x 10^5)]
x ≈ 0.025 m
Therefore, the spring should be compressed approximately 0.025 meters (or 2.5 cm) to launch the satellite with the given specifications.
(2) To check if the force required to stop the hand and launch pad after launch exceeds the motor's limit, we can calculate the force using the concept of momentum.
Given:
Distance to stop (d) = 20 cm = 0.2 m
Mass of the hand and launch pad (m1 + m2) = 1550 kg
The final velocity of the hand and launch pad (v_final) is 0 m/s (since they need to stop).
The initial momentum (p_initial) is:
p_initial = (m1 + m2) x v
The final momentum (p_final) is:
p_final = 0 (since v_final = 0)
According to the law of conservation of momentum:
p_initial = p_final
Therefore:
(m1 + m2) x v = 0
Now we can solve for v:
v = 0 / (m1 + m2)
v = 0
Since the velocity is 0, there is no kinetic energy involved and all the energy must come from the motor's force.
The work done by the motor's force can be calculated as:
W = F x d
W = Force x distance
Force = W / d
Given that the distance is 0.2 m and the work done is equal to the change in potential energy (since kinetic energy is 0), we can find the required force:
Potential energy (U) = (1/2) k x^2
Change in potential energy = U_initial - U_final
Change in potential energy = (1/2) k x^2 - 0
Since the potential energy is given as 9.7% less than the initial stored energy, we have:
Change in potential energy = 0.903 x (1/2) k x^2
Therefore:
W = 0.903 x (1/2) k x^2
Substituting the known values:
Force = (0.903 x (1/2) k x^2) / d
Force = (0.903 x (1/2) x (1.7 x 10^5 N/m) x (0.025 m)^2) / 0.2 m
Calculating the force:
Force ≈ 4143 N
The force required to stop the hand and launch pad is approximately 4143 N. Since this force is less than the motor's design limit of 2.4 x 10^4 N, the motor will not be exceeded.
To determine how far the spring should be compressed, we can use the principle of conservation of energy. The energy stored in the spring when it is compressed will be equal to the kinetic energy of the satellite when it is launched.
Step 1: Calculate the spring potential energy.
The potential energy stored in a spring can be given by:
Potential energy = 0.5 * k * x^2
where k is the spring constant and x is the compression of the spring.
Step 2: Calculate the kinetic energy of the satellite.
The kinetic energy of an object can be given by:
Kinetic energy = 0.5 * m * v^2
where m is the mass of the satellite and v is the velocity at which it is launched.
Step 3: Equate the potential energy to the kinetic energy and solve for x.
0.5 * k * x^2 = 0.5 * m * v^2
Substituting the given values:
k = 1.7 x 10^5 N/m
m = 300 kg
v = 2 m/s
0.5 * (1.7 x 10^5) * x^2 = 0.5 * 300 * (2^2)
Simplifying the equation:
1.7 x 10^5 * x^2 = 300 * 4
x^2 = (300 * 4) / (1.7 x 10^5)
x^2 = 0.0071
Taking the square root of both sides:
x = sqrt(0.0071)
x ≈ 0.084 m (rounded to three decimal places)
Therefore, the spring should be compressed to approximately 0.084 meters.
Now, let's move on to checking the force required to stop the hand plus launch pad after the launch. Using the principle of conservation of momentum, the force exerted by the motor can be calculated.
Step 1: Calculate the initial momentum of the system.
The initial momentum is given by:
Initial momentum = (mass of hand + mass of launch pad) * initial velocity
Step 2: Calculate the final momentum of the system.
The final momentum is given by:
Final momentum = (mass of hand + mass of launch pad) * final velocity
Since the hand stops after a distance of 20 cm, the final velocity is zero.
Step 3: Calculate the change in momentum.
The change in momentum is given by:
Change in momentum = Final momentum - Initial momentum
Step 4: Calculate the force exerted by the motor.
The force exerted by the motor to stop the hand can be given by:
Force = Change in momentum / time taken to stop
Since the time taken to stop is not provided, we cannot calculate the exact force required. However, we can check if the force required exceeds the designed limit of 2.4 x 10^4 N.
If the force required exceeds 2.4 x 10^4 N, the motor is not suitable for this application.
Note: To calculate the time taken to stop, additional information such as the deceleration rate of the motor or the braking distance may be required.
To check how far the spring should be compressed, we need to determine the amount of potential energy stored in the spring. We can use the formula for potential energy stored in a spring:
Potential Energy = (1/2) * k * x^2
where k is the spring constant (1.7 x 10^5 N/m) and x is the compression distance. We want to find the compression distance that ensures the spring loses 9.7% of its stored energy as thermal energy.
Let's denote the total potential energy stored in the spring as ET, and the thermal energy as Eth. We can set up the equation:
(Eth / ET) * 100% = 9.7%
Now, to solve for x, we can rearrange the equation by substituting the formulas for ET and Eth:
(Eth / (0.5 * k * x^2)) * 100% = 9.7%
Simplifying further:
Eth = (0.097 * 0.5 * k * x^2)
Now we have an equation to find the thermal energy Eth based on the compression distance x. We can substitute the given values:
Eth = (0.097 * 0.5 * 1.7 x 10^5 N/m * x^2)
To check if the force required to stop the hand plus launch pad (due to recoil) exceeds the motor's design limit (2.4 x 10^4 N), we need to calculate the recoil force.
The recoil force can be determined using Newton's second law of motion:
Force = mass * acceleration
The mass we need to consider is the total mass of the hand (350 kg) plus the launch pad (1200 kg) plus the small satellite (300 kg).
Now, let's calculate the acceleration. We know that the hand stops after traveling a distance of 20 cm, which is equal to 0.2 meters. We can use the formula for deceleration:
Deceleration = (final velocity^2 - initial velocity^2) / (2 * distance)
Given that the final velocity is 0 m/s (the hand stops) and the initial velocity is 2 m/s (as measured by the International Space Station), we can calculate the deceleration.
Once we have the deceleration, we can calculate the recoil force by multiplying the total mass by the deceleration.
Let's go ahead and perform these calculations.