University Physics I · topic UPI-04.03
Newton's second law
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Newton's Second Law: predict a finite change
Newton's second law says that, in an inertial frame, the vector sum of the external forces on a body equals the product of its inertial mass and its acceleration, so a = ΣF/m and the acceleration points along the net force. Use the supplied topic model Y_out = hypot(5+f2*cosd(th), f2*sind(th))/m, where Y_out is ACCELERATION |a| (m s⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; hypot(a,b) means √(a²+b²); sind/cosd/tand use degrees. m is Puck mass m, measured in kg. Keep f2 (Size of second force F₂) = 4 N; th (Angle θ between F₁ and F₂) = 60 ° fixed. For this exercise use only 1.2125 kg ≤ m ≤ 2.7875 kg on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at m = 1.559 kg and 1.7795 kg. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δm across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ACCELERATION |a| against m
Predictions from the supplied Newton's Second Law model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ACCELERATION |a| / m s⁻²
Horizontal axis (x): m: Puck mass m / kg
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Puck mass m / kg | ACCELERATION |a| / m s⁻² |
|---|---|
| 1.2125 | 6.441443031675593 |
| 1.6062499999999997 | 4.862412249591693 |
| 1.9999999999999998 | 3.9051248379533283 |
| 2.39375 | 3.2627674886294127 |
| 2.7874999999999996 | 2.8018832918050784 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of m separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At m = 1.559 kg, substitute into Y_out = hypot(5+f2*cosd(th), f2*sind(th))/m: Y₀ = 5.00978 m s⁻².
- At m = 1.7795 kg, the same fixed-parameter model gives Y₁ = 4.38901 m s⁻².
- Signed change: ΔY_out = Y₁ − Y₀ = -0.620768 m s⁻²; the input change is Δm = 0.2205 kg.
- Average sensitivity is ΔY_out/Δm = -2.81527 m s⁻² per kg. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 5.00978 m s⁻², Y₁ = 4.38901 m s⁻²; ΔY_out = -0.620768 m s⁻²; average sensitivity = -2.81527 m s⁻² per kg.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Newton's Second Law: infer an input with uncertainty
Newton's second law says that, in an inertial frame, the vector sum of the external forces on a body equals the product of its inertial mass and its acceleration, so a = ΣF/m and the acceleration points along the net force. Use the supplied topic model Y_out = hypot(5+f2*cosd(th), f2*sind(th))/m, where Y_out is ACCELERATION |a| (m s⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; hypot(a,b) means √(a²+b²); sind/cosd/tand use degrees. m is Puck mass m, measured in kg. Keep f2 (Size of second force F₂) = 4 N; th (Angle θ between F₁ and F₂) = 60 ° fixed. For this exercise use only 1.2125 kg ≤ m ≤ 2.7875 kg on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 3.593811 m s⁻², with a bounded reading uncertainty ±0.11879 m s⁻². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 kg, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
ACCELERATION |a| against m
Predictions from the supplied Newton's Second Law model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ACCELERATION |a| / m s⁻²
Horizontal axis (x): m: Puck mass m / kg
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Puck mass m / kg | ACCELERATION |a| / m s⁻² |
|---|---|
| 1.2125 | 6.441443031675593 |
| 1.6062499999999997 | 4.862412249591693 |
| 1.9999999999999998 | 3.9051248379533283 |
| 2.39375 | 3.2627674886294127 |
| 2.7874999999999996 | 2.8018832918050784 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(m) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(1.2125) = 6.44144 m s⁻² and Y_out(2.7875) = 2.80188 m s⁻². The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives m_line = 1.2125 + (3.59381 − 6.44144)(2.7875 − 1.2125)/(2.80188 − 6.44144) = 2.4448 kg. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives m = 2.17325 kg; substituting back gives Y_out = 3.593811 m s⁻².
- The interpolation error in the input is |m_line − m| = 0.271548 kg. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 3.475021 m s⁻² and 3.712601 m s⁻². Sort the roots even if the response decreases: 2.10371 kg ≤ m ≤ 2.24754 kg.
- The lower and upper input deviations are 0.0695363 kg and 0.0742903 kg. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer m ≈ 2.17325 kg; compatible interval [2.10371, 2.24754] kg. Endpoint interpolation gives 2.4448 kg.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.