Q: A 10-kg ladder of length $$L$$ leans against a frictionless wall at an angle of $$60^\circ$$ with the ground. What is the minimum coefficient of static friction $$\mu_s$$ required at the ground to prevent slipping?

$$\mu_s = \frac{1}{2} \tan 60^\circ$$

Q: A participant in a strongman competition uses a 4-meter beam to lift a 9000 N load suspended 0.85 m from the pivot. If the man lifts the beam at its end, 3.6 m from the pivot, what force must he apply to hold the beam steady?

$$F = \frac{9000 \times 0.85}{3.6} \text{ N}$$

Q: A spring stretches from 10 cm to 22 cm when a force of 100 N is applied. What is the spring constant $$k$$?

$$k = \frac{100}{0.12} \text{ N/m}$$

Q: A car has a track width $$t$$ and its center of gravity is at height $$h. $$What is the critical angle $$\theta_c at $$which the car will tip over?

$$\theta_c = \tan$$^{-1}$$ \left( \frac{t}{h} \right)$$

Q: A rod of length $$L$$ and area $$A is $$stretched by a force $$F$$ and the change in length is $$\Delta L. $$What is the tensile stress in the rod?

$$\text{Stress} = \frac{F}{A}$$

Question 1

A solid cylinder of mass $$M$$ and radius $$R is $$released from rest at the top of a ramp and rolls without slipping down the incline. What is the relationship between the cylinder's linear velocity $$v$$ and its angular velocity $$\omega as it $$reaches the bottom?

Accepted Answer

$$v = \omega R$$

Question 2

Which of the following statements about static equilibrium is correct?

Accepted Answer

An object is in static equilibrium if both the net force and the net torque on it are zero.

Question 3

A 10-kg ladder of length $$L$$ leans against a frictionless wall at an angle of $$60^\circ$$ with the ground. What is the minimum coefficient of static friction $$\mu_s$$ required at the ground to prevent slipping?

Accepted Answer

$$\mu_s = \frac{1}{2} 	an 60^\circ$$

Question 4

A participant in a strongman competition uses a 4-meter beam to lift a 9000 N load suspended 0.85 m from the pivot. If the man lifts the beam at its end, 3.6 m from the pivot, what force must he apply to hold the beam steady?

Accepted Answer

$$F = \frac{9000 	imes 0.85}{3.6} 	ext{ N}$$

Question 5

A spring stretches from 10 cm to 22 cm when a force of 100 N is applied. What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{100}{0.12} 	ext{ N/m}$$

Question 6

Which of the following best describes the moment of inertia $$I$$ for a system of point masses $$m_1$$, $$m_2$$, $$m_3 at $$distances $$r_1$$, $$r_2$$, $$r_3$$ from the axis?

Accepted Answer

$$I = m_1$ r_1$^2 + m_2$ r_2$^2 + m_3$ r_3$^2$$

Question 7

A car has a track width $$t$$ and its center of gravity is at height $$h. $$What is the critical angle $$	heta_c at $$which the car will tip over?

Accepted Answer

$$	heta_c = 	an$$^{-1}$$ \left( \frac{t}{h} ight)$$

Question 8

A steel rod of length $$L$$ and cross-sectional area $$A is $$stretched by a force $$F. $$What is the extension $$\Delta L of $$the rod, given Young's modulus $$Y$$?

Accepted Answer

$$\Delta L = \frac{F L}{A Y}$$

Question 9

Which of the following is true about the torque produced by a force $$F$$ applied at a distance $$r$$ from a pivot at an angle $$\phi$$?

Accepted Answer

$$	au = r F \sin \phi$$

Question 10

A bucket of mass $$m is $$attached to a rope wound around a cylinder of mass $$M$$ and radius $$R. If $$the bucket is released from rest and the rope does not slip, what is the acceleration $$a of $$the bucket?

Accepted Answer

$$a = \frac{g}{1 + \frac{M}{m}}$$

Question 11

Which of the following materials has the highest Young's modulus according to the provided data?

Accepted Answer

Steel

Question 12

A rod is stretched by a force $$F$$ and returns to its original length when the force is removed. What region of the stress-strain curve is this behavior associated with?

Accepted Answer

Elastic region

Question 13

A wheel rolls without slipping on a flat surface. What is the velocity of the point at the top of the wheel relative to the ground?

Accepted Answer

$$v_{top} = 2 v_{center}$$

Question 14

Which of the following best describes Hooke's law for a spring?

Accepted Answer

$$F = -kx$$

Question 15

If the center of gravity of an object moves outside its base of support, what happens?

Accepted Answer

The object becomes unstable and may tip over.

Question 16

A rope is wrapped around a pulley of radius $$R. If $$the pulley rotates with angular acceleration $$\alpha$$, what is the linear acceleration $$a of $$the rope?

Accepted Answer

$$a = \alpha R$$

Question 17

Which of the following is a correct statement about the choice of pivot point when calculating torque for static equilibrium?

Accepted Answer

Any point can be chosen as the pivot; the net torque will be zero at all points in static equilibrium.

Question 18

A biological material has a Young's modulus of $$0.15 	imes 10^9$ N/m^2$. Which material is this most likely to be?

Accepted Answer

Tendon

Question 19

A rod of length $$L$$ and area $$A is $$stretched by a force $$F$$ and the change in length is $$\Delta L. $$What is the tensile stress in the rod?

Accepted Answer

$$	ext{Stress} = \frac{F}{A}$$

Rotational Motion, Static Equilibrium, and Elasticity: Study Notes for Physics for Life Sciences I (Lecture 13)

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