A bunch of MasteringPhysics problems that I don't understand

[adamc637]

SHM with torque? Springs and frequency mass relation?

Problem 1

A slender, uniform, metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod, with the other end of the spring attached to a rigid support.

If the rod is displaced by a small angle Theta from the vertical and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle Theta is small enough for the approximations $${\rm sin} \Theta \approx \Theta$$ and $${\rm cos} \Theta \approx 1$$ to be valid. The motion is simple harmonic if $$d^{2} \theta /dt^{2}= - \omega ^{2} \theta$$ , and the period is then $$T=2 \pi / \omega .$$)

The answer is $$2\pi\sqrt{\frac {M}{3k}}$$

So $$T = 2\pi\sqrt{\frac{m}{k}}$$ usually right? So how do I calculate the period when the spring is giving the force?

Do I use $$\tau = -kx*r$$? But where is r?

Do I need to use the physical pendulum equation and use the 1/12MR^2 equation? But once again, I don't have R. I'm confused on even where to start!

Problem 2:
A partridge of mass 5.10 kg is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down 0.100 m below its equilibrium position and released, it vibrates with a period of 4.17 s.

I got the speed at equilibrium position (.151 m/s), and the acceleration at .05m above equilibrium (-.113 m/s^2).

This is where I got stuck:
When it is moving upward, how much time is required for it to move from a point 0.050 m below its equilibrium position to a point 0.050 m above it?

The acceleration varies, so do I have to find some type of integral? Or maybe do I take some ratio of the period? I have no idea!

The motion of the partridge is stopped, and then it is removed from the spring. How much does the spring shorten?

Ummm, if the spring is of negligible mass, how do I calculate the amount the spring will shorten? What formula do I use? Argh! This oscillation concept is killing me!

Problem 3:

The scale of a spring balance reading from zero to 200 N is 12.5 cm long. A fish hanging from the bottom of the spring oscillates vertically at 2.60 Hz.

What is the mass of the fish? You can ignore the mass of the spring.

I drew a free-body diagram with $$F_{spring} = m_{fish}*g$$

So the period is .3846 s, and $$k = \frac {m}{(\frac {T}{2\pi})^2}$$.

I tried to find k from some other formula, but I don't know what x to use for $$F = -kx$$ and I have no idea what the 0 to 200 N has to do with this problem. I'm confused again...

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Sorry for posting so much easy problems here, but I don't think my mind is working correctly lately :yuck:. Thanks!

Adam

 

[adamc637]

I missed my TA's office hours!

 

[thepatientmental]

It's completely normal to struggle with mastering physics problems, especially when they involve advanced concepts like SHM with torque and springs. Here are some tips that might help you understand these problems better:

1. Start by understanding the concept: Before attempting to solve the problem, make sure you understand the concept behind it. For example, in the first problem, understand that SHM (Simple Harmonic Motion) is a type of motion where the restoring force is directly proportional to the displacement from equilibrium. Also, understand that torque is a rotational force that causes angular acceleration.

2. Use the given approximations: As the hint suggests, for small angles, you can use the approximations sinΘ ≈ Θ and cosΘ ≈ 1. These approximations will simplify the equations and make it easier for you to solve the problem.

3. Use the equations for SHM: For the first problem, you can use the equation d^2θ/dt^2 = -ω^2θ to solve for the angular frequency ω. Once you have ω, you can use the equation T = 2π/ω to find the period.

4. Use the equation for torque: In the first problem, you can use the equation τ = -kx*r to find the torque, where x is the displacement from equilibrium and r is the distance from the pivot point. This equation relates the torque to the force exerted by the spring.

5. Use the equation for physical pendulum: In some cases, you may need to use the equation for the physical pendulum, which is T = 2π√(I/mgd), where I is the moment of inertia, m is the mass, g is the acceleration due to gravity, and d is the distance from the pivot point.

6. Use the equation for spring constant: In the second problem, you can use the equation k = m(2π/T)^2 to find the spring constant, where T is the period and m is the mass of the fish. This equation relates the spring constant to the period of oscillation.

7. Draw a free-body diagram: It's always helpful to draw a free-body diagram to visualize the forces acting on the object. This will help you set up the correct equations and solve the problem more accurately.

8. Practice, practice, practice: The more you practice, the better you will become at solving these types of problems. Don't be